Misunderstanding basic math concepts, help please?
Moderators: gmalivuk, Moderators General, Prelates

 Posts: 450
 Joined: Wed Sep 21, 2011 3:44 am UTC
Re: Misunderstanding basic math concepts, help please?
The problem is that you're taking the law of the excluded middle, which says essentially that every sentence is either true or false with no 'middle' option, and you're trying to equate that law to something completely different. You're trying to equate it to some law about things being 100% unambiguous. But the law of the excluded middle deals with the nature of truth, while the concept of ambiguity is about knowledge.
If I say to to you "The gate current of a MOSFET tends to be lower than the base current of a bipolar transistor", I can't be sure whether you've understood what I said, but regardless, what I said is either true or false. "Ambiguity" is mostly about that first part: the question about whether you understand what I'm saying. The law of the excluded middle is about that second part: the fact that regardless of your understanding, what I said is either true or false.
In terms of "ambiguity", everything is always a continuum, because you can never be 100% sure that someone understands what you're trying to say.
When developing mathematics from scratch, the first foundation is human language. Because whatever basic ideas we might start with, we have to communicate those ideas to each other. It is at this step that the "ambiguity" of mathematics is a continuum, and one of the goals is to start out with ideas that are very simple so that we can be as confident as possible that we all understand those ideas.
Regarding what those first ideas are, you're essentially right that we start with laws of logical thought, and typically(but not necessarily) we include the law of the excluded middle in this. But it's important to remember that that comes after we take it for granted that informal human language works well enough to express the idea.
If I say to to you "The gate current of a MOSFET tends to be lower than the base current of a bipolar transistor", I can't be sure whether you've understood what I said, but regardless, what I said is either true or false. "Ambiguity" is mostly about that first part: the question about whether you understand what I'm saying. The law of the excluded middle is about that second part: the fact that regardless of your understanding, what I said is either true or false.
In terms of "ambiguity", everything is always a continuum, because you can never be 100% sure that someone understands what you're trying to say.
When developing mathematics from scratch, the first foundation is human language. Because whatever basic ideas we might start with, we have to communicate those ideas to each other. It is at this step that the "ambiguity" of mathematics is a continuum, and one of the goals is to start out with ideas that are very simple so that we can be as confident as possible that we all understand those ideas.
Regarding what those first ideas are, you're essentially right that we start with laws of logical thought, and typically(but not necessarily) we include the law of the excluded middle in this. But it's important to remember that that comes after we take it for granted that informal human language works well enough to express the idea.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Before we get on to more complicated logic...
No. Your law of the excluded middle question is already asking about MUCH more complicated logic than the problems I asked you to work on. So
Before we get on to more complicated logic...
Propositional logic worksheet:
http://www.luc.edu/faculty/avande1/logi ... oblems.pdf
Rules of propositional logic:
http://www.luc.edu/faculty/avande1/logi ... rules.pdf
Please try to answer 2 of the problems on each page of the worksheet. If you can't answer any of those problems or don't know what to do you can ask questions about it and I can help you answer. This worksheet is the simplest form of deductive logic that people work with. The questions you are asking are about subject matter much deeper than the subject of the worksheet I have presented, so therefore, it makes sense to make sure everyone understands the worksheet more before we delve into more complicated topics.
It's to your merit that you are interested in these deep questions, but at some point we need to get our heads out of the theoretical clouds and actually get our hands dirty doing some ACTUAL examples of what we are talking about. A lot can be learned very quickly that way.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5493
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Lexington, MA
 Contact:
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:at some point we need to get our heads out of the theoretical clouds and actually get our hands dirty doing some ACTUAL examples of what we are talking about. A lot can be learned very quickly that way.
This is exceedingly true, but also illustrates a very generous bar for what we're calling dirt.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Law of excluded middle, wiki wrote:In logic, the law of excluded middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is true.
Before we get on to more complicated logic... it appears I can't read this simple passage properly.
I was under the impression that mathematics is built upon thought. That the laws of thought are the laws of mathematics. Am I mistaken in this?
Arbiterofthruth reminds us of the exception that proves the rules. There is indeed a branch of mathematics that explicitly subverts the law of the excluded middle.
If mathematics is founded in something other than thought as defined by the three classic laws; that could go a long way to explaining where I'm going wrong.
What is the basis for mathematical thought? How does this differ from that specified by the three classic laws?
Wikipedia was going on the more philosophical side of logic, while here we are talking about formal logic. What Wikipedia mentioned was irrelevant to our current discussion. What we are talking about is a very concrete axiom (scheme) present in certain logic systems.
The words "laws of thought" or "three classical laws" mean nothing our this context. We are (I assume) considering formal logic as a set of formal rules of manipulating symbols. No philosophy is allowed to enter. There is no socalled "basis" for mathematical thought.
Twistar wrote:Before we get on to more complicated logic...
Propositional logic worksheet:
http://www.luc.edu/faculty/avande1/logi ... oblems.pdf
Rules of propositional logic:
http://www.luc.edu/faculty/avande1/logi ... rules.pdf
Please try to answer 2 of the problems on each page of the worksheet. If you can't answer any of those problems or don't know what to do you can ask questions about it and I can help you answer. This worksheet is the simplest form of deductive logic that people work with. The questions you are asking are about subject matter much deeper than the subject of the worksheet I have presented, so therefore, it makes sense to make sure everyone understands the worksheet more before we delve into more complicated topics.
It's to your merit that you are interested in these deep questions, but at some point we need to get our heads out of the theoretical clouds and actually get our hands dirty doing some ACTUAL examples of what we are talking about. A lot can be learned very quickly that way.
While it is important that Treatid actually understands logic, I would have to say that this is a very unfair request, because the questions assume some deduction system presented by the relevant notes/course materials, which is impossible for us to figure out, and how we can approach the questions depend a lot on the actual deduction system used.
To make things clear, here are some things from actual formal logic I copied out of my lecture notes.
We start by defining the language, which are the things we are linguistically allowed to write down. This is not really important in our discussion, so you can skip over this (spoilered) section if you are not bothered.
Spoiler:
Now we can talk about "proving" things. We start with the axioms and deduction rules of our system. Note that we are just naming these sentences as axioms/deduction rules. We are not claiming they are "true", "valid" or anything. They just are. Also, at least for the purposes of this discussion, there is no reason why we chose these axioms in particular. It just happens that these are quite short axioms that happen to be rather useful.
Our system of deduction composes of the following three axioms:
 p ⇒ (q ⇒ p)
 [p ⇒ (q ⇒ r)] ⇒ [(p ⇒ q) ⇒ (p ⇒ r)]
 (¬¬p) ⇒ p
A proof of a statement t is a finite sequence t_{1}, t_{2}, ..., t_{n} of propositions, with t_{n} = t, such that each t_{i} is one of the following:
 An axiom
 A proposition t_{i} such that there exist j, k < i with t_{j} being t_{k}⇒ t_{i}. This is known as modus ponens, abbreviated MP.
Example proof:
We want to prove p ⇒ p.
 [p ⇒ ((p ⇒ p) ⇒ p)] ⇒ [(p ⇒ (p ⇒ p)) ⇒ (p ⇒ p)] (Axiom 2)
 p ⇒ ((p ⇒ p) ⇒ p) (Axiom 1)
 [p ⇒ (p ⇒ p)] ⇒ (p ⇒ p) (MP on 1, 2)
 p ⇒ (p ⇒ p) (Axiom 1)
 p ⇒ p (MP on 3, 4)
Of course, this is a rather minimal deduction system, and we wouldn't be able to do, say, set theory with this. However, coming up with a more complicated system is just a matter of adding a few more symbols and a few more axioms. It is fundamentally the same. This is it.
To say a system is inconsistent, we mean there is a proof of ⊥ in the system. The only possible vagueness, if existent at all, is the ability of the reader to understand English and determine whether a list of propositions is in fact a proof.
When we say rejecting the law of excluded middle, we simply mean not including (¬¬p) ⇒ p in the list of axioms, so that there are fewer things we can prove.
Re: Misunderstanding basic math concepts, help please?
dalcde wrote:While it is important that Treatid actually understands logic, I would have to say that this is a very unfair request, because the questions assume some deduction system presented by the relevant notes/course materials, which is impossible for us to figure out, and how we can approach the questions depend a lot on the actual deduction system used.
It's not an unfair request and it's not ambiguous. I posted the associated rules of deduction and even gave an example of what I'm looking for. Anyone who has taken a class in formal logic would be able to complete my request. Whether Treatid has taken such a course or not, he acts as if he already knows everything that one would learn in one of those classes so I'm trying to get him to demonstrate that knowledge so that we can find some more common ground.
You can be overly pedantic* and say I haven't defined my language blah blah blah, but at the level of discourse we're having here it's pretty clear what you're supposed to do. Furthermore, it's these exact steps that are causing so much confusion for Treatid so I was trying SPECIFICALLY to avoid all of the type of stuff you brought up in your post.
I was trying to ask for something very simple and I worry that your post just obfuscated everything again by bringing the discussion back to the exact topics that Treatid struggles with and I have a strong hunch he's going to latch on to all of that with his usual confusion and conflation and once again ignore my request for him to demonstrate a simple task that any beginner logic student could complete.
*I'm not tossing aside the importance of properly and rigorously defining language. Obviously that is critical and in a different conversation I would not call it overly pedantic. However, in response to the simple request of doing a few line proof, it is pedantic to ask for all of that infrastructure. I supplied what was necessary to complete the task.
Re: Misunderstanding basic math concepts, help please?
Much as I imagine it is tempting to just assume that everything I say is wrong, the classic laws of thought are explicitly the foundation of mathematical logic and all of mathematics that does not explicitly disavow the specified law. This information can be readily found through links on the Law of excluded middle, wiki.
However consciously aware any given mathematician is, it is a fundamental assumption of all mathematics that all propositions are binary, true/false, excluded middle, unless specifically stated otherwise. I guess it is one of those things that is so deeply embedded you forget it is there at all.
(I didn't know that the excluded middle was such a fundamental concept in mathematics at the beginning of this thread, or that it would appear to be the key to the difference between the way I see axiomatic mathematics and everyone else sees it. Thank you for helping me get here).
It looks to me like some of the points being raised are to do with the idea of the excluded middle and its place in mathematics. These are standard, possibly even fundamental, parts of mathematics. I don't think I'm the right person to be defending or justifying the status quo.
I think it helps to understand why such a choice was made at such a fundamental level of mathematics:
A proposition could have:
1. Just two possible states: True or nottrue.
2. Many possible states: True, nottrue and things that are either 'neither true or nottrue' or 'both true and nottrue'.
3. Many possible states: things that are neither true or nottrue (true and false not being valid states).
Pick one.
(see also the classic law of thought regarding identity).
...
Twistar feels that my ability to logic is in doubt, despite this thread generally agreeing with the form of the deductions I've presented but criticising my assumptions and/or context for any given deduction (not that known assumptions and context aren't critical elements of logic).
Question 1, page 1 of the worksheet that twistar provided (and reference sheet):
Premise 1: D ⊃ E
Premise 2: E ⊃ F
Premise 3: F ⊃ G
Show: D ⊃ G
Prem1 to prem 2 via hyp. sill.
Prem2 to prem 3 via hyp. sill.
done. (Turns out the first question is a bit on the trivial side...)
Now, take a moment to consider (D ⊃ E). What is the alternative to (D ⊃ E)?
~(D ⊃ E), of course.
Logic doesn't deal with the question of "D sortofbutnotactuallyimplies E". Natural languages; the English language deals with nonbinary propositions all the time. Mathematical propositions are universally x vs !x rather than x, !x and something that is a bit of both (or neither) (*exceptions may apply).
@Arbiteroftruth: You are wholly right. I do equate "100% unambiguous" with the law of the excluded middle. I think of "100% unambiguous" as being a definite thing. Either a thing is completely (100% unambiguous) or it is not(100% unambiguous).
Pedantically, this applies to pretty much everything: "My milk jug either is, or is not, 1/3 full". Not entirely coincidentally, the Number Line is typically composed of points representing individual numbers rather than... whatever the alternative would be.
As mathematicians, a number is a discrete entity. It is either the number, or it is not the number.
As humans, we find a bit of wiggle room useful. sorta, kinda, ish, nearenough, are essential modifiers to real world experience.
Mathematics explicitly starts out by disavowing the possibility of wriggle room. Applying wriggle room to mathematics is applying English to Mathematics  all the rigour and logic we went to a lot of trouble to build up drowns in a mountain of wriggle room.
However consciously aware any given mathematician is, it is a fundamental assumption of all mathematics that all propositions are binary, true/false, excluded middle, unless specifically stated otherwise. I guess it is one of those things that is so deeply embedded you forget it is there at all.
(I didn't know that the excluded middle was such a fundamental concept in mathematics at the beginning of this thread, or that it would appear to be the key to the difference between the way I see axiomatic mathematics and everyone else sees it. Thank you for helping me get here).
It looks to me like some of the points being raised are to do with the idea of the excluded middle and its place in mathematics. These are standard, possibly even fundamental, parts of mathematics. I don't think I'm the right person to be defending or justifying the status quo.
I think it helps to understand why such a choice was made at such a fundamental level of mathematics:
A proposition could have:
1. Just two possible states: True or nottrue.
2. Many possible states: True, nottrue and things that are either 'neither true or nottrue' or 'both true and nottrue'.
3. Many possible states: things that are neither true or nottrue (true and false not being valid states).
Pick one.
(see also the classic law of thought regarding identity).
...
Twistar feels that my ability to logic is in doubt, despite this thread generally agreeing with the form of the deductions I've presented but criticising my assumptions and/or context for any given deduction (not that known assumptions and context aren't critical elements of logic).
Question 1, page 1 of the worksheet that twistar provided (and reference sheet):
Premise 1: D ⊃ E
Premise 2: E ⊃ F
Premise 3: F ⊃ G
Show: D ⊃ G
Prem1 to prem 2 via hyp. sill.
Prem2 to prem 3 via hyp. sill.
done. (Turns out the first question is a bit on the trivial side...)
Now, take a moment to consider (D ⊃ E). What is the alternative to (D ⊃ E)?
~(D ⊃ E), of course.
Logic doesn't deal with the question of "D sortofbutnotactuallyimplies E". Natural languages; the English language deals with nonbinary propositions all the time. Mathematical propositions are universally x vs !x rather than x, !x and something that is a bit of both (or neither) (*exceptions may apply).
@Arbiteroftruth: You are wholly right. I do equate "100% unambiguous" with the law of the excluded middle. I think of "100% unambiguous" as being a definite thing. Either a thing is completely (100% unambiguous) or it is not(100% unambiguous).
Pedantically, this applies to pretty much everything: "My milk jug either is, or is not, 1/3 full". Not entirely coincidentally, the Number Line is typically composed of points representing individual numbers rather than... whatever the alternative would be.
As mathematicians, a number is a discrete entity. It is either the number, or it is not the number.
As humans, we find a bit of wiggle room useful. sorta, kinda, ish, nearenough, are essential modifiers to real world experience.
Mathematics explicitly starts out by disavowing the possibility of wriggle room. Applying wriggle room to mathematics is applying English to Mathematics  all the rigour and logic we went to a lot of trouble to build up drowns in a mountain of wriggle room.
Re: Misunderstanding basic math concepts, help please?
Wiggle room is useful in real life because we don't have perfect information  and, more importantly, we know we don't have perfect information.
Maths comes from a different direction: It says 'if A is true and 'A implies B' is true, then we can deduce that B is true'. It bypasses the fact we don't have perfect knowledge by saying 'but what if we did?'
And that turns out to be really useful because it seems like the universe operates on mathematical lines. So the better our imperfect knowledge becomes, the better we seem to be able to predict how the universe behaves.
Maths comes from a different direction: It says 'if A is true and 'A implies B' is true, then we can deduce that B is true'. It bypasses the fact we don't have perfect knowledge by saying 'but what if we did?'
And that turns out to be really useful because it seems like the universe operates on mathematical lines. So the better our imperfect knowledge becomes, the better we seem to be able to predict how the universe behaves.
Re: Misunderstanding basic math concepts, help please?
Godel pointed out some issues with the Law of the Excluded Middle  or, rather, he took some issues that had been known about since ancient Greece, and showed that they applied more generally than people realised.
The Liar Paradox (along with a whole bunch of other logical paradoxes) gives you what appears to be a statement, but which cannot consistently be either true or false.
What Godel showed is that versions of the Liar Paradox exist in systems where you wouldn't expect them  such as standard arithmetic.
There are also many things in mathematics to which the concepts of "true" and "false" do not apply. Is "7" true or false?
And the fact that some mathematics does explicitly set aside Excluded Middle means that, while it is a default assumption, it's not a necessary part of mathematics  otherwise, the stuff that excludes it would not be mathematics.
The Liar Paradox (along with a whole bunch of other logical paradoxes) gives you what appears to be a statement, but which cannot consistently be either true or false.
What Godel showed is that versions of the Liar Paradox exist in systems where you wouldn't expect them  such as standard arithmetic.
There are also many things in mathematics to which the concepts of "true" and "false" do not apply. Is "7" true or false?
And the fact that some mathematics does explicitly set aside Excluded Middle means that, while it is a default assumption, it's not a necessary part of mathematics  otherwise, the stuff that excludes it would not be mathematics.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:A proposition could have:
1. Just two possible states: True or nottrue.
2. Many possible states: True, nottrue and things that are either 'neither true or nottrue' or 'both true and nottrue'.
3. Many possible states: things that are neither true or nottrue (true and false not being valid states).
Pick one.
Putting a formalist hat on, I disagree with this. A proposition is either provable or not provable. It is neither true, nor false, nor anything inbetween. If a proposion is not provable, it might be provable with more assumptions. If a proposition is provable, it might not be provable with fewer assumptions (eg. without the law of excluded middle).
If we want to introduce truth, it is certainly conceivable that there are many possible "truth values". It just depends on what system you pick.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:...
Twistar feels that my ability to logic is in doubt, despite this thread generally agreeing with the form of the deductions I've presented but criticising my assumptions and/or context for any given deduction (not that known assumptions and context aren't critical elements of logic).
Question 1, page 1 of the worksheet that twistar provided (and reference sheet):
Premise 1: D ⊃ E
Premise 2: E ⊃ F
Premise 3: F ⊃ G
Show: D ⊃ G
Prem1 to prem 2 via hyp. sill.
Prem2 to prem 3 via hyp. sill.
done. (Turns out the first question is a bit on the trivial side...)
This doesn't convince me of your ability to do formal logic. If this was a quiz and the problems were out of 4 I would give this a 2/4. Even though the question is trivial you didn't answer it in the right form.
Here's what would have looked better.
1._____D ⊃ E_____Prem
2._____E ⊃ F_____Prem
3._____F ⊃ G_____Prem_____Show: D ⊃ G
4._____D ⊃ F_____1,2 Hyp. Syll.
5._____D ⊃ G_____3,4 Hyp. Syll.
Let me explain why I wrote the answer out this way and why your answer is incorrect. Furthermore, this is EXTREMELY critical. The way you wrote your answer is a MUCH more ambiguous derivation than me. If you think your answer is 100% correct then it is no wonder you are worried that logic is ambiguous.
Here is the structure of a logical proof. There are 3 columns and a variable number of rows.
The 1st column just has the number of the line we are working on.
The 2nd column has a sentence written in whatever language we are working in. It is necessary that this sentence follow the rules of syntax of the given language.
The 3rd column is the justification for writing down the sentence which appears in the 2nd column. The is the meat of the proof.
Notice that for this particular proof the 3rd column has a couple of different things. For the three premises it just says premise. This means we don't need prior justification for writing down those sentences. "premise" is enough justification. However, if you look at line 4 you see that this is a derived sentence because it doesn't say "premise" in column 3, instead it says "1,2 hyp. syll." This means that line 4 was derived FROM lines 1 and 2 using the rule of hypothetical syllogism or "hyp. syll." as it appears on the reference sheet. This line is notable because it is derived ONLY from the premises. However, if you look at the next line, line 5, you see that the this line was derived from lines 3 and 4 again via "hyp. syll." This one is notable because it uses one premise but it ALSO uses the derived formula. If the proof was longer eventually we would get lines which are justified only by derived sentences, but of course it could all be traced back to the premises. Notice that the sentence on line 5 matches the sentence next to "Show:". This means that the proof is complete.
Anyways, we move forward in this fashion because we want to be EXTREMELY particular about how we are using each premise, and each sentence to move to the next one. On each step of the derivation we write which lines were necessary to derive that line and what rule was used to derive it.
You're answer is ambiguous:
Prem1 to prem 2 via hyp. sill.
Prem2 to prem 3 via hyp. sill.
done. (Turns out the first question is a bit on the trivial side...)
Prem 1 to prem 2 via hyp. syll.? That doesn't make sense. It sounds to me like you are trying to say that premise 2 is derived from premise 1 via hyp. syll. But that is not correct. Both Premise 1 and premise 2 are premises so they aren't derived from anything. They are "given". Then you say prem2 to prem 3 via hyp. syll. It sounds again like you are saying premise 3 is derived from premise 2 now but that doesn't make sense. It's the same mistake as before.
The reason you get 2/4 instead of 0/4 is because your intuition was right that this proof relies on the use of hyp. syll. two times, and that you will need to apply it (in some form) to all of the premises. However, it was only your intuition that was correct. your formalism was woefully lacking. And I am not being nitpicky here. Formalism is the essence of formal logic. If you don't get the formalism right you are doomed to fail and misunderstand things.
edit3: Also, you can't argue back that you had the right idea and I was just misunderstanding how you said it. Clarity of communication is the ENTIRE purpose of formal logic. So if you fail do that the fault is with you and not with me.
Try answering #3 on the worksheet. I'm sorry if this feels slow and frustrating. I promise that once you show you can answer these sorts of problems proficiently we can talk about the law of the excluded middle.
worksheet
reference sheet
edit: Formatted the proof to look nicer.
edit2: Easy link to my proof of #4 for another example of how to do a proof:
Last edited by Twistar on Mon May 23, 2016 5:11 pm UTC, edited 2 times in total.

 Posts: 450
 Joined: Wed Sep 21, 2011 3:44 am UTC
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:@Arbiteroftruth: You are wholly right. I do equate "100% unambiguous" with the law of the excluded middle. I think of "100% unambiguous" as being a definite thing. Either a thing is completely (100% unambiguous) or it is not(100% unambiguous).
The first half of this quote isn't saying the same thing as the second half. The law of the excluded middle doesn't say anything about things being 100% unambiguous. It does say that the *sentence* "this is 100% unambiguous" is either true or false (applying the law of the excluded middle informally to informal language, rather than in formal logic). The law of the excluded middle says something about that *sentence* about ambiguity, but it doesn't say anything about ambiguity itself.
That's why we can use the law of the excluded middle and at the same time say that nothing is ever 100% unambiguous. For anything you might say, the thing you said either is or is not 100% unambiguous. More specifically, for anything you might say, the thing you said definitely is not 100% unambiguous. Everything anyone ever says fails to be 100% unambiguous, and that's consistent with the law of the excluded middle.
But that just shows that "is this 100% unambiguous" is a useless question to ask. A more useful question is something like "is this at least 90% unambiguous". Again, by the law of the excluded middle, everything either is or is not at least 90% unambiguous. "Ambiguity" is still a continuum, even though the law of the excluded middle means that statements *about* ambiguity are either true or false. That's what you're missing with the way you're trying to equate the concepts.
The law of the excluded middle is a statement about statements. "Ambiguity" is not a statement, so it's not something that the law of the excluded middle is talking about.
Re: Misunderstanding basic math concepts, help please?
You know that feeling when you aren't sure whether a comment is serious or sarcastic (poe's law)?
In axiomatic mathematics 'provable' and 'true' are the same thing. If it can be proven (with respect to the axioms) then it is true (with respect to the axioms).
It looks to me like a lot of effort is being expended on anything but the point at hand.
Now that I know for certain that all mathematical propositions relating to axiomatic mathematics are excluded middle (unless explicitly noted otherwise); I need to find out whether this is as damaging to axiomatic mathematics as I think it is.
The first thing to note about divided middle propositions is that they don't have a middle bit. For every mathematical proposition* there is no semix.
The idea of 99% knowledge is not a mathematical proposition. This applies to every element of an excluded middle system. Each element of a proposition must also be excluded middle in nature (x, !x) else the proposition cannot be resolved. Excluded middle is exclusive of included middle. One or the other. Not both.
I think this has been the source of some of the friction in this thread. I've been encouraged to consider real world examples, or natural languages (English) as counterpoints, in this thread. However, when making mathematical propositions about axiomatic mathematics, we are only permitted excluded middle propositions. It is either x or !x. Partiallyx is not a meaningful concept in mathematics.
"There is no try. Do. Or do not."
Jumping back the the beginning of this thread: A thing either is an axiomatic system or it is not. Nonaxiomaticthings are completely nonaxiomaticthings. Partiallyaxiomatic does not compute.
There is perfection or there is nothing. As far as axiomatic mathematics is concerned, either axioms, rules and symbols are specified with complete and utter certainty, or they are not specified at all. Where axiomatic mathematics is concerned, one, tiny, insignificant seeming, lack of complete and utter certitude in any one part means that we are utterly uncertain about everything.
Regarding whatever it is that axiomatic mathematics does, there is not a middle ground. Either every single element is known and understood in every conceivable detail... or nothing is known about anything. Those are the only two options.
Which brings us back to these 3 choices:
1. Just two possible states: True or nottrue.
2. Many possible states: True, nottrue and things that are either 'neither true or nottrue' or 'both true and nottrue'.
3. Many possible states: things that are neither true or nottrue (true and false not being valid states).
We, tried choice 1. Turns out it was a mistake. Choice 2 is just silly. This leaves choice 3  the one that real life and English illustrate with such ease.
...
Twistar: If you cannot point to a specific fault in my deductions then that means there is no fault in my deductions (probably). It doesn't mean that I'm failing at logic or using some weird, Treatid only, logic.
My guess is that you are currently in academia and see tests and exams as the ground state of being. Here in the semireal world, this thread is an actual chain of deduction. Either the deductions follow the established rules or they don't. Either you can see an actual fault in the application of rules of deduction or you can't.
Furthermore, you shouldn't believe what I say because you think I'm good at logic so anything I say will be reasonable. You should look at the actual deductions on display.
I could be a gibbering monkey randomly typing (it is what monkeys do...) and accidentally type out a unification of General Relativity and Quantum Mechanics. The source doesn't matter  is the deduction, as laid out, valid?
Whether I'm generally good or generally bad at logic doesn't change whether this particular deduction is well formed or not.
*All mathematical propositions except where explicitly stated otherwise
In axiomatic mathematics 'provable' and 'true' are the same thing. If it can be proven (with respect to the axioms) then it is true (with respect to the axioms).
It looks to me like a lot of effort is being expended on anything but the point at hand.
Now that I know for certain that all mathematical propositions relating to axiomatic mathematics are excluded middle (unless explicitly noted otherwise); I need to find out whether this is as damaging to axiomatic mathematics as I think it is.
The first thing to note about divided middle propositions is that they don't have a middle bit. For every mathematical proposition* there is no semix.
The idea of 99% knowledge is not a mathematical proposition. This applies to every element of an excluded middle system. Each element of a proposition must also be excluded middle in nature (x, !x) else the proposition cannot be resolved. Excluded middle is exclusive of included middle. One or the other. Not both.
I think this has been the source of some of the friction in this thread. I've been encouraged to consider real world examples, or natural languages (English) as counterpoints, in this thread. However, when making mathematical propositions about axiomatic mathematics, we are only permitted excluded middle propositions. It is either x or !x. Partiallyx is not a meaningful concept in mathematics.
"There is no try. Do. Or do not."
Jumping back the the beginning of this thread: A thing either is an axiomatic system or it is not. Nonaxiomaticthings are completely nonaxiomaticthings. Partiallyaxiomatic does not compute.
There is perfection or there is nothing. As far as axiomatic mathematics is concerned, either axioms, rules and symbols are specified with complete and utter certainty, or they are not specified at all. Where axiomatic mathematics is concerned, one, tiny, insignificant seeming, lack of complete and utter certitude in any one part means that we are utterly uncertain about everything.
Regarding whatever it is that axiomatic mathematics does, there is not a middle ground. Either every single element is known and understood in every conceivable detail... or nothing is known about anything. Those are the only two options.
Which brings us back to these 3 choices:
1. Just two possible states: True or nottrue.
2. Many possible states: True, nottrue and things that are either 'neither true or nottrue' or 'both true and nottrue'.
3. Many possible states: things that are neither true or nottrue (true and false not being valid states).
We, tried choice 1. Turns out it was a mistake. Choice 2 is just silly. This leaves choice 3  the one that real life and English illustrate with such ease.
...
Twistar: If you cannot point to a specific fault in my deductions then that means there is no fault in my deductions (probably). It doesn't mean that I'm failing at logic or using some weird, Treatid only, logic.
My guess is that you are currently in academia and see tests and exams as the ground state of being. Here in the semireal world, this thread is an actual chain of deduction. Either the deductions follow the established rules or they don't. Either you can see an actual fault in the application of rules of deduction or you can't.
Furthermore, you shouldn't believe what I say because you think I'm good at logic so anything I say will be reasonable. You should look at the actual deductions on display.
I could be a gibbering monkey randomly typing (it is what monkeys do...) and accidentally type out a unification of General Relativity and Quantum Mechanics. The source doesn't matter  is the deduction, as laid out, valid?
Whether I'm generally good or generally bad at logic doesn't change whether this particular deduction is well formed or not.
*All mathematical propositions except where explicitly stated otherwise
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:In axiomatic mathematics 'provable' and 'true' are the same thing. If it can be proven (with respect to the axioms) then it is true (with respect to the axioms).
No.
ETA: provability is a welldefined concept, namely that there is a proof. There are many ways we can define truth. If we define "true" to mean "valid in all models satisfying the axioms", then saying "provable" and "true" are the same thing corresponds to the completeness of the theory, which is not always true, and if true, is a theorem. If we believe that there is some external notion of truth, then provability is definitely not the same thing as truth. If we define "true" to mean "provable", then your statement is contentless.

 Posts: 450
 Joined: Wed Sep 21, 2011 3:44 am UTC
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:In axiomatic mathematics 'provable' and 'true' are the same thing.
No they aren't. The fact that they aren't is essentially the point of Godel's theorems.
Treatid wrote:It looks to me like a lot of effort is being expended on anything but the point at hand.
The point at hand, as per the thread title, is that you don't understand basic math concepts and supposedly want help. People are avoiding the points you're trying to discuss because your comprehension of those points is marred by deeper misunderstandings that need to be addressed first. You were doing reasonably well early in the thread, in terms of actually listening to what people have to say. Now you've partially grasped one or two things you didn't understand before, assumed that was all you needed to fix your misunderstandings, and returned to your modus operandi of ignoring what people are saying and insisting you have found some profound truth, when really you just still have a lot of fundamental misunderstandings.
Treatid wrote:Jumping back the the beginning of this thread: A thing either is an axiomatic system or it is not. Nonaxiomaticthings are completely nonaxiomaticthings. Partiallyaxiomatic does not compute.
True, but your understanding of what "axiomatic" means is still fatally flawed. It is in this understanding of what "axiomatic" means in the first place where nonblackandwhite notions are useful.
Treatid wrote:There is perfection or there is nothing. As far as axiomatic mathematics is concerned, either axioms, rules and symbols are specified with complete and utter certainty, or they are not specified at all.
This is wrong, for reasons related to things we discussed before about languages vs. theories. But until you come back around to sincerely listening and wanting help, it would be pointless to go into more detail than that. My point here is that this isn't a question being dodged; it's a question being set aside until your more fundamental misunderstandings are resolved well enough that you can grasp the answer.
Treatid wrote:Twistar: If you cannot point to a specific fault in my deductions then that means there is no fault in my deductions (probably). It doesn't mean that I'm failing at logic or using some weird, Treatid only, logic.
He did point to specific faults in your 'deductions'. First, that you didn't actually write down any statements in the language in formal logic, and thus didn't make any deductions at all. Second, that the most natural translation of your informal deductions into formal logic results in an invalid deduction that doesn't even reach the desired conclusion.
And this is important, because it's symptomatic of your big issues. You're using informal reasoning to make grand claims about the nature of formal reasoning. But these grand claims are the result of 1) faulty reasoning, and 2) misunderstanding of formal logic. A better understanding of formal logic would not only eliminate much of your misunderstanding, but it would also better equip you to recognize the flaws in your informal reasoning.
Do the exercises. You came to this thread asking for help. At this point I think Twistar is right, and you need to start at the rote mechanical level before we'll even be on the same page about how people actually use math(in the sense of what they physically write down on paper). Once you're familiar with that, we can return to talking about languages and theories and truth and how all those things relate to this practice of physically writing symbols on paper. But your misunderstandings seem severe enough that first we have to make sure you understand that basic practice before we can go further.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:The first thing to note about divided middle propositions is that they don't have a middle bit. For every mathematical proposition* there is no semix.
The law of the excluded middle is about truth values, not propositions.
For any proposition x, either x is true or !x is true, but never both. For example, let x be the proposition "all odd numbers are prime". x is false, and !x is true. The law of the excluded middle has been obeyed. Note that !x says "it is not the case that not all odds are prime". !x does NOT say "there are no odd primes".
Can "semix" propositions exist? Certainly. Here is one: "some odd numbers are prime". This is a totally coherent mathematical sentence, which happens to be true (and its negation is false).
Here's another: "all odd numbers that end with 7 are prime". Again, coherent mathematical sentence, but it happens to be false, and its negation is true.
The law of the excluded middle only talks about the truth values you are allowed to have. It doesn't say anything about what kind of propositions you are allowed to make.
I say this because, while you seem to get this basically right this when talking about a generic proposition x, you jump right to "therefore there are no odd primes" territory with stuff like this:
Treatid wrote:Jumping back the the beginning of this thread: A thing either is an axiomatic system or it is not. Nonaxiomaticthings are completely nonaxiomaticthings. Partiallyaxiomatic does not compute.
There is perfection or there is nothing. As far as axiomatic mathematics is concerned, either axioms, rules and symbols are specified with complete and utter certainty, or they are not specified at all.
The law of the excluded middle only tells you that there is perfection, or there is not perfection. If things are not specified perfectly, then they may still be specified imperfectly; you cannot conclude that they are not specified at all. This is a completely unjustified, "get an F on your Logic 101 homework"level error.
I honestly can't tell if you're confused about the law of the excluded middle, or confused at a more basic level about negations in general.
No, even in theory, you cannot build a rocket more massive than the visible universe.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Now that I know for certain that all mathematical propositions relating to axiomatic mathematics are excluded middle (unless explicitly noted otherwise); I need to find out whether this is as damaging to axiomatic mathematics as I think it is.
I hear you loud and clear, Treatid. You've given up pretending to be looking for help now that you've baited people in and have returned to full crackpot mode. It was a clever ruse, and it worked quite well: you exploited the natural didactic tendencies of people in this forum. Did you have the whole thing planned from the outset, or did you come across the law of the excluded middle during the topic and realize you could twist it to your own ends?
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Twistar: If you cannot point to a specific fault in my deductions then that means there is no fault in my deductions (probably). It doesn't mean that I'm failing at logic or using some weird, Treatid only, logic.
My guess is that you are currently in academia and see tests and exams as the ground state of being. Here in the semireal world, this thread is an actual chain of deduction. Either the deductions follow the established rules or they don't. Either you can see an actual fault in the application of rules of deduction or you can't.
Furthermore, you shouldn't believe what I say because you think I'm good at logic so anything I say will be reasonable. You should look at the actual deductions on display.
I could be a gibbering monkey randomly typing (it is what monkeys do...) and accidentally type out a unification of General Relativity and Quantum Mechanics. The source doesn't matter  is the deduction, as laid out, valid?
Whether I'm generally good or generally bad at logic doesn't change whether this particular deduction is well formed or not.
I'm not going to waste my time arguing with you anymore. "Failing at logic or using some weird, Treatid only, logic" is the best description anyone has ever given of what you have been doing for over a year.
My only role in this thread from here on out will be to encourage you to solve problems on the worksheet so that we can confirm we are all talking about the same thing when we talk about logic. I encourage you to solve problem 3 on the worksheet. Maybe to spark your interest I'll point out that the law of the excluded middle appears as the first statement under Logical Truths on the reference sheet. Isn't that interesting? Like I said before, as soon as you show you can perform these simple deductions in a satisfactory way, I promise we can talk about the law of the excluded middle. If you're struggling to answer the worksheet problem you should feel free to ask questions in this thread. I and others would be more than happy to help. Here's a video on Youtube that seems ok which might help as well.
worksheet
reference sheet
Furthermore, I encourage everyone in this thread to stop responding to Treatid until he shows us that he commands a basic understanding of deduction in formal logic.
And Treatid, a note to you. You are very passionate and interested in logic, clearly. If you are really so interested you should be excited to learn more about it, it's confusing to me that you aren't interested to learn. Unfortunately my only interpretation of your behavior is that it stems from a bit of arrogance and stubbornness. You want so badly to be that person who has an idea that no one has thought of before that you have blinded yourself to the basics. This is so unfortunate because, even though people (and academics whom you apparently dislike) have already thought of the basics, they are still very interesting and can lead to very interesting ideas and conversations. Much more exciting and interesting than the discussion we are having now.
You have to learn to walk before you can run.
Reassess what your goals are with this pursuit of formal logic. Me asking you to solve a simple deduction exercise really isn't a tall order. You should be able to see that. Good luck.
Re: Misunderstanding basic math concepts, help please?
Cauchy is correct. My last post went a little crackpotty. I jumped from "all mathematical propositions regarding axiomatic mathematics are excluded middle" through "axiomatic mathematics demands perfection, anything less than perfection is nothing" to "so axiomatic mathematics is dead. ok. What's next?".
My basis for seeing doom and gloom for a purely excluded middle axiomatic mathematics is the following:
Consider some excluded middle system X who's purpose is to 'describe' all z. 'Describe' is just a label, only the most abstract properties as relates to an excluded middle system are important.
X > 'describe' all z.
Assuming we want X to be 'describable' then I think the following are valid deductions:
1. X must be a member of z.
2. All 'describe' are members of z.
3. All z are 'described' within X.
4. notz cannot be 'described' within X.
5. This is a closed loop (Nothing from outside X can impinge on anything inside X).
6. Every 'description' requires a 'description' in order to be 'described'.
7. All chains of 'describing' 'descriptions' must be circular.
Do these deductions follow from the premise?
...
@Meteoric: Your illustrations appear to be of collections. Even where the proposition refers to some odd numbers, each of the individual numbers is definitely odd or even, prime or not prime.
The whole point and purpose of The Law of Excluded Middle is that the middle is explicitly excluded. There is no middle ground between 'specified' and 'notspecified'. "specified imperfectly" has been deliberately removed from the table as a possibility.
@Twistar: I apologise for any offence. I have nothing against academics. I've been one myself. To the extent that I know you, you seem a pretty decent chap.
However, you are coming across as if you have just discovered formal logic and all must now bow down before your great font of wisdom.
You've gone from questioning how well I understand formal logic to stating that whatever I do, it has no connection to logic.
For whatever reason, it looks like you are not acting in good faith.
As such, it seems easier for everything I say to be examined for logical flaws, than for me to obtain the certificate of "Never, ever makes logical flaws under any circumstances... 100% guaranteed!"
So, thank you, I formally decline your invitation to jump through your hoops. Best of luck with the Boycott.
My basis for seeing doom and gloom for a purely excluded middle axiomatic mathematics is the following:
Consider some excluded middle system X who's purpose is to 'describe' all z. 'Describe' is just a label, only the most abstract properties as relates to an excluded middle system are important.
X > 'describe' all z.
Assuming we want X to be 'describable' then I think the following are valid deductions:
1. X must be a member of z.
2. All 'describe' are members of z.
3. All z are 'described' within X.
4. notz cannot be 'described' within X.
5. This is a closed loop (Nothing from outside X can impinge on anything inside X).
6. Every 'description' requires a 'description' in order to be 'described'.
7. All chains of 'describing' 'descriptions' must be circular.
Do these deductions follow from the premise?
...
@Meteoric: Your illustrations appear to be of collections. Even where the proposition refers to some odd numbers, each of the individual numbers is definitely odd or even, prime or not prime.
Meteoric wrote:The law of the excluded middle only tells you that there is perfection, or there is not perfection. If things are not specified perfectly, then they may still be specified imperfectly; you cannot conclude that they are not specified at all. This is a completely unjustified, "get an F on your Logic 101 homework"level error.
The whole point and purpose of The Law of Excluded Middle is that the middle is explicitly excluded. There is no middle ground between 'specified' and 'notspecified'. "specified imperfectly" has been deliberately removed from the table as a possibility.
@Twistar: I apologise for any offence. I have nothing against academics. I've been one myself. To the extent that I know you, you seem a pretty decent chap.
However, you are coming across as if you have just discovered formal logic and all must now bow down before your great font of wisdom.
You've gone from questioning how well I understand formal logic to stating that whatever I do, it has no connection to logic.
For whatever reason, it looks like you are not acting in good faith.
As such, it seems easier for everything I say to be examined for logical flaws, than for me to obtain the certificate of "Never, ever makes logical flaws under any circumstances... 100% guaranteed!"
So, thank you, I formally decline your invitation to jump through your hoops. Best of luck with the Boycott.

 Posts: 450
 Joined: Wed Sep 21, 2011 3:44 am UTC
Re: Misunderstanding basic math concepts, help please?
Treatid, the first half of your post is, frankly, incoherent. So I'm going to skip it.
That last sentence does not follow from the one before it. "Specified imperfectly" is very much on the table; it's just that it gets classified as either "specified" or "not specified". The law of the excluded middle would say that a thing is either "specified" or "not specified", but the law of the excluded middle has nothing to say about what impact the adjective "imperfectly" has on that classification.
The fallacy you're committing here is one of equivocation. You can informally use the law of the excluded middle to say that everything is either "specified" or "not specified", and you can use the law of the excluded middle to say that everything is either "perfectly specified" or "not perfectly specified", but you're not doing either of those. You're mixing and matching those two examples to come up with "perfectly specified" or "not specified". You don't get to drop the adjective "perfectly" on a whim; logic is more strict than that.
This is why you need to do the exercises Twistar posted. People aren't kidding when they tell you you have no idea what you're talking about. Do the exercises and familiarize yourself with how logic works on a mechanical level. If you really do want help, then that's the best place you can start.
Treatid wrote:The whole point and purpose of The Law of Excluded Middle is that the middle is explicitly excluded. There is no middle ground between 'specified' and 'notspecified'. "specified imperfectly" has been deliberately removed from the table as a possibility.
That last sentence does not follow from the one before it. "Specified imperfectly" is very much on the table; it's just that it gets classified as either "specified" or "not specified". The law of the excluded middle would say that a thing is either "specified" or "not specified", but the law of the excluded middle has nothing to say about what impact the adjective "imperfectly" has on that classification.
The fallacy you're committing here is one of equivocation. You can informally use the law of the excluded middle to say that everything is either "specified" or "not specified", and you can use the law of the excluded middle to say that everything is either "perfectly specified" or "not perfectly specified", but you're not doing either of those. You're mixing and matching those two examples to come up with "perfectly specified" or "not specified". You don't get to drop the adjective "perfectly" on a whim; logic is more strict than that.
This is why you need to do the exercises Twistar posted. People aren't kidding when they tell you you have no idea what you're talking about. Do the exercises and familiarize yourself with how logic works on a mechanical level. If you really do want help, then that's the best place you can start.
Re: Misunderstanding basic math concepts, help please?
Just so that we know what we are talking about, could Treatid please
 Describe formally some particular axiomatic system (eg. Peano Arithmetic, including the background logic system, say, first order logic of some sort)
 State precisely the law of excluded middle in that axiomatic system.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:The whole point and purpose of The Law of Excluded Middle is that the middle is explicitly excluded. There is no middle ground between 'specified' and 'notspecified'. "specified imperfectly" has been deliberately removed from the table as a possibility.
For the thousandth time, the Law of the Excluded Middle relates to truth values. Specifically, it says that for any given statement, either it's true, or its negation is true. (I mean, read the wiki article you linked to, it's right there in the first sentence.) So either "a thing is specified" is true, or "not(a thing is specified)" is true. You seem to have claimed that "not(a thing is specified)" is the same thing as "a thing is notspecified". Depending on what you mean by "notspecified", this may or may not be the case. Does "specified imperfectly" fall under "specified"? Or does it fall under "notspecified"? If the former, then you haven't removed "imperfectly specified". If the latter, then you haven't removed "imperfectly specified". If neither, then "not(a thing is specified)" and "a thing is notspecified" aren't actually the same, because upon imperfect specification, the former statement returns true, and the latter statement returns false. No matter what, you haven't "removed ['specified imperfectly'] from the table", and the law of the excluded middle will never be able to do so.
You're somewhere on the spectrum between "does not understand the law of the excluded middle" and "does not understand negation", and it's hard to tell where because your posts use a lot of wishywashy terms and phrasing. (Seriously, what does "Consider some excluded middle system X who's purpose is to 'describe' all z." mean?) This is why people want you to do these exercises and whatnot: it's because we can't actually tell what you do and don't know. Your showing on the worksheet question was, quite frankly, subpar. It was correct only under very, very generous interpretations of what you wrote, so it still leaves us with the question: Does Treatid actually know basic logic, or are we merely tricking ourselves into believing that he knows basic logic? People are really suspecting the latter, and that's where the boycott is coming from: if you don't know basic logic and are unreceptive to learning it, then you can't reasonably expect people to have a rational discourse with you. If you unequivocally demonstrated that you know basic logic, it would go a long way towards helping your case. This involves getting the correct answer, yes, but you also need to write the explanation clearly, so that there's no doubt that the thing you meant is the correct thing.
Treatid wrote:As such, it seems easier for everything I say to be examined for logical flaws, than for me to obtain the certificate of "Never, ever makes logical flaws under any circumstances... 100% guaranteed!"
If your arguments have logical flaws, why should we accept them?
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Re: Misunderstanding basic math concepts, help please?
Cauchy wrote:If your arguments have logical flaws, why should we accept them?
Umm...
Personally, if a deduction has logical flaws, I might consider pointing out those logical flaws so that the deducer can learn from their errors.
I can't imagine why any one would continue to consider an argument to be valid after (critical) logical flaws have been identified within the argument.
If you assume that everything I say is gibberish  then everything I say will look like gibberish to you.
However, the rest of your post is a well reasoned, on point, critical response to my argument that you say is impossible to understand.
To that point:
I agree with your characterisation of my point. I do see 'describable' and not'describable' as simple opposites with no possible middle ground.
I think I understand your view. Not(perfectly'describable') allows for (imperfectly'describable'). And, technically, I agree with this.
However, (perfectly'describable') and (imperfectly'describable') seem to be entirely distinct concepts. There is no way to plug (imperfectly'describable') into a (perfectly'describable') system. There is no path to (perfectly'describable'). There is no mechanism for (imperfectly'describable) to morph into (perfectly'describable').
For all practical purposes (perfectly'describable') and (imperfectly'describable') appear to be binary opposites. A thing is one or the other, not both.
It looks to me like the label has changed but the excluded middle remains.
...
I specify a particular excluded middle system X to 'describe' all z.
X > 'describe' all z.
Since I want X to be 'describable' then I think the following are valid deductions:
1. X must be a member of z.
2. All 'describe' are members of z.
3. All z are 'described' within X.
4. notz cannot be 'described' within X.
5. This is a closed loop (Nothing from outside X can impinge on anything inside X).
6. Every 'description' requires a 'description' in order to be 'described'.
7. All chains of 'describing' 'descriptions' must be circular.
Do these deductions follow from the premise?
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:I specify a particular excluded middle system X to 'describe' all z.
X > 'describe' all z.
Since I want X to be 'describable' then I think the following are valid deductions:
1. X must be a member of z.
2. All 'describe' are members of z.
3. All z are 'described' within X.
4. notz cannot be 'described' within X.
5. This is a closed loop (Nothing from outside X can impinge on anything inside X).
6. Every 'description' requires a 'description' in order to be 'described'.
7. All chains of 'describing' 'descriptions' must be circular.
Do these deductions follow from the premise?
No. Half of these are wrong and the other half are too incoherent to even be wrong.
Is Peano Arithmetic a member of the natural numbers (#1)? Does geometry's ability to describe triangles hinge on its inability to describe circles (#4)? Is all the talk of "undefined terms" in math or logic textbooks a smokescreen (#6)?
No, even in theory, you cannot build a rocket more massive than the visible universe.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Personally, if a deduction has logical flaws, I might consider pointing out those logical flaws so that the deducer can learn from their errors.
People have done that constantly with you. That's what Twistar was doing when he "graded" your response to the worksheet question. You seemed... unwilling to learn from your errors there, which was troubling to quite a few people.
If you assume that everything I say is gibberish  then everything I say will look like gibberish to you.
It's not that people were assuming that what you said was gibberish. When people here say that what you write is incoherent, that's not (just) a dismissal: it's a request for clarification. If you can't accurately your ideas to other people, then it doesn't really matter whether they're actually logically sound or anything, because no one else will be able to tell. In fact, I'd say that people are assuming that what you say isn't gibberish. They're just deriving contradictions from that and concluding that the opposite must actually be true.
However, the rest of your post is a well reasoned, on point, critical response to my argument that you say is impossible to understand.
I took a stab at one paragraph, one of the few paragraphs where I could tell what you were talking about. Don't mistake my critique for an understanding of your entire argument.
However, (perfectly'describable') and (imperfectly'describable') seem to be entirely distinct concepts.
When you say "distinct", do you mean "have no overlap", or do you mean "unrelated to each other in derivation"? Because I'd agree with the former, but not with the latter.
There is no way to plug (imperfectly'describable') into a (perfectly'describable') system.
What? I don't know what it means to plug an adjective into a system.
There is no path to (perfectly'describable').
What do you mean, path?
There is no mechanism for (imperfectly'describable) to morph into (perfectly'describable').
I don't know how adjectives morph.
See, this is what I was talking about. I'm following along with your argument okay, and then you start using strange turns of phrase that I can't decipher. Now I have no idea what you're talking about. I can either give up on the whole thing, or I can ignore this part and hope that I can figure out what you meant to say through context in the later parts.
For all practical purposes (perfectly'describable') and (imperfectly'describable') appear to be binary opposites. A thing is one or the other, not both.
Through the context of this line, I believe I know what you were trying to say in the previous paragraph. You're trying to say that something can't be both perfectlydescribable and imperfectlydescribable, and that there's no temporal component to a things describability. That is, if something could be perfectly described at some later date, then it's perfectlydescribable at all times (including now) and we merely lack the ingenuity to come up with a perfect description. I agree with you on that.
For all practical purposes (perfectly'describable') and (imperfectly'describable') appear to be binary opposites. A thing is one or the other, not both.
(Quoted again because I'm addressing it directly this time) You've established that a thing cannot be both perfectlydescribable and imperfectlydescribable, sure. But I disagree with your claim that a thing must be one or the other. Imperfectlydescribable means, based on the words it comprises, that a thing can be described in some fashion, but that every such description is not perfect. Or rather, if you want it to mean something other than that, then you ought to say what that meaning is. In any case, under what I take to be the standard meaning of imperfectlydescribable, then you've still left the door open to things which are notatalldescribable, that is, things for which there exists no description. I'd tell you about them, but I can't describe them. You need more argument to show that such a thing doesn't exist (which is tantamount to saying that every thing has at least one description).
It looks to me like the label has changed but the excluded middle remains.
You haven't established that yet, though that's not really the salient problem here. Rather, your use of the phrase "excluded middle" is very weird here, because, for the thousand and first time, the Law of the Excluded Middle relates to truth values. It states that for a given proposition, either it's true, or its negation is true. If you define an adjective A, then for any given thing, one of "the thing is A" or "not(the thing is A)" is true. And if you define notA by a thing being notA precisely when "not(the thing is A)", then yes, for every thing, either it's A or it's notA. That's a corollary of the Law of the Excluded Middle, to be sure, but it doesn't mean that the Law of the Excluded Middle suddenly has an adjectival mode, or that finding an A/notA pair makes your system an Excluded Middle system (whatever that even means).
Why are you so fixated on the term "excluded middle"? Your continued use of it makes your arguments sound very crackpotish, as though you think the Law of the Excluded Middle says much more than it actually does, and that systems that have the Law of the Excluded Middle as an axiom have some sort of mystic significance.
I specify a particular excluded middle system X to 'describe' all z.
I have to stop you right here, because I don't know what this means.
You have an "excluded middle system", X. Is an "excluded middle system" one in which the Law of the Excluded Middle is an axiom? Why give this a special name? Looking down at the rest of your argument, it never seems to come up that X is an "excluded middle system", so why bother specifying that at all?
This system " 'describes' all z". I really, really don't know what this means. You've got a word in quotes, so I assume that your use of the word describe is not a standard one (since that's what people typically mean when they use quotes around a single word in the fashion you have). But you don't supply this definition anywhere else. So I can't reasonably evaluate any of your claims about things being described.
You also say "all z". What is z here? My mind immediately jumps to one possibility. In the same fashion as one might say "all integers", z here represents a collection of things you already have, and X is a system that describes them (which is to say, you have a model of them in X, I guess; as I said above, it's unclear what describe means when you use it, so I'm giving a favorable interpretation). But if that's the case, your first deduction is ludicrous: why on earth would it follow that X itself must be a member of z? That's a nonsequitur.
So I guess my interpretation of "all z" is wrong. Maybe you meant that z is the collection of all things that are describable? (Again, it's vague what that even means at this point.) X is a system that gives a description to all of the describable things. But why should such a system even exist? This seems to reek of the same naivete that led directly to Russell's Paradox in set theory. If I can make a system that describes any collection of things that I can talk about, then why can't I make a system Y that describes precisely those systems that don't describe themselves? Does Y describe itself? Without consulting any rules about what things systems can and can't describe (and indeed, without even supplying a working definition of describe), you're rather hardpressed to assert the existence of this system X.
Or did you mean something else by "all z"? I could only come with two interpretations that I thought were remotely reasonable, but maybe I merely missed yours. Without your expounding on the meanings of what you write, it's hard for us to even get started on the accuracy of the argument you set forth, because any critique of the argument either relies on a possibly incorrect interpretation of the things you've written, or else ignores the parts that require strong interpretation and instead attacks the parts that seem wrong under a variety of interpretations. I believe Meteoric has taken the second tack, and more generally this is why most effort is "being expended on anything but the point at hand": it's because no one can figure out what you mean when you talk about the point at hand.
Incidentally, if you were using more standard terminology, this would be much less of an issue. The reason terminology like this gets standardized is so that people can use it without having to explain in detail what they mean by each term they use. When you eschew the standard terminology and start using your own, you also have to explain what you mean by your terms, or risk people not being able to understand what you mean when you write things.
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:However, you are coming across as if you have just discovered formal logic and all must now bow down before your great font of wisdom.
No. Rather, it is that I know what formal logic is and I know that what you are doing is not formal logic. So no, I'm not demanding that you bow down. I'm just demanding that you follow the rules of formal logic. It's pretty tough because you have no idea what those rules are.
If you think you know what those rules are let us know.
If you think you don't know what those rules are you should let us know too.
If you do think you know what the rules of formal logic are how about you tell us?
You've gone from questioning how well I understand formal logic to stating that whatever I do, it has no connection to logic.
Until you prove to me that you can do a simple deduction in formal logic I am going to assume that you do not know how to do simple deductions in formal logic. The reason I'm ignoring all of the crap you're spouting about the excluded middle is because it is not possible to have an intelligent conversation about the excluded middle unless all parties involved know how to do simple deductions in formal logic.
Do you agree or disagree with the statement that it's impossible to have an intelligent conversation about the excluded middle unless all parties involved know how to do simple deductions in formal logic?
For whatever reason, it looks like you are not acting in good faith.
Just trying to teach you about what you asked about.
Please solve number three in the format I've demonstrated and described above:
worksheet
reference sheet
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:The whole point and purpose of The Law of Excluded Middle is that the middle is explicitly excluded. There is no middle ground between 'specified' and 'notspecified'. "specified imperfectly" has been deliberately removed from the table as a possibility.
Just going to comment on the excluded middle thing here. As others have said (repeatedly), excluded middle has nothing to do with specificity. It has to do with truth. Consider the statement:
All dogs go to heaven.
Excluded middle says that this statement is either true or false. So either all dogs go to heaven, or at least one dog does not go to heaven. That's all it can tell you. It doesn't matter how well or poorly defined the terms are. Whatever the terms "dogs" and "heaven" are said to mean, there is a definite truth value to the statement. Note that the excluded middle does not imply that either "All dogs go to heaven" or "No dogs go to heaven" must be true. The negation of "All dogs to go heaven" is simply "It is not the case that all dogs go to heaven", or, more simply "At least one dog does not go to heaven".
In terms of specificity, we can look at it the same way "All statements in axiomatic mathematics are perfectly specified." Excluded middle again says that this statement is true or false. If it is false, then the following must be true "At least one statement in axiomatic mathematics is imperfectly specified." We cannot conclude from this that "No statements in axiomatic mathematics are perfectly specified."
A statement can be true or false. Excluded middle says nothing about the precision of a statement. A statement that is unspecific, say, "If some stuff happens then other stuff will happen" still has some truth value to it, it's just that the truth of the statement might not be meaningful in practical terms if the terms are too poorly defined.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:Spoiler:
Twistar's requests have been reasonable, perceived tone aside. All this nitpicking is necessary to make sure everyone is talking about the same thing, especially because you seem to be coming to different logical conclusions than other people in this thread. This could be because either you are wrong, someone else here is wrong, or there is just a failure to communicate clearly (or some combination of these). Surely if you and everyone else were correct and able to understand each other, then we wouldn't be spinning our wheels here, no?
The point of the worksheets are to make sure you can 1) demonstrate that you know how formal logic works, and 2) express your thoughts to us in a way that we all can understand. The point isn't to insult your intelligence. This foundational math stuff isn't exactly common knowledge to all but the most diehard of math people. Heck, this stuff wasn't introduced to me until grad school, and I am still not an expert on the subject by any means. If you really want to learn, you have to put the work in. Otherwise you are just wasting your time, as well as everyone else's.
Re: Misunderstanding basic math concepts, help please?
How about in a show of good faith everyone in the thread answers one of the worksheet problems? Just to prove that it's a common language we can all use. I've already done two so I won't do another.
Re: Misunderstanding basic math concepts, help please?
Sure, Twistar. I'll take a crack at #14.
Edit: You're absolutely right, Cauchy. Not skipping steps is hard when dealing with this kind of material!
Spoiler:
Edit: You're absolutely right, Cauchy. Not skipping steps is hard when dealing with this kind of material!
Last edited by cyanyoshi on Tue May 31, 2016 5:33 pm UTC, edited 2 times in total.
Re: Misunderstanding basic math concepts, help please?
I'll take #8.
I'd also nitpick both of the previous solutions.
For #14,
For #6,
Spoiler:
I'd also nitpick both of the previous solutions.
For #14,
Spoiler:
For #6,
Spoiler:
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Re: Misunderstanding basic math concepts, help please?
@Cauchy: Thank you very, very, much for your recent responses. They are well reasoned, coherent and tothepoint. I hugely respect the amount of effort you are putting into trying to understand me, despite myself.
Much of what you are saying I agree with. For example, Your careful analysis of my attempt to communicate a set of premises and logical deductions clearly explained where I was being ambiguous or straight up unclear. I hope my followup attempt fixes these issues.
There are points where I don't fully agree with you  but even here you have explained your position, shown how it differs from mine and presented a good argument to justify that view. I will, of course, argue these points (ideally, 'argue': to discuss until consensus is reached; as opposed to blindly championing a cause). I see that your points are well founded and that compelling evidence needs to be shown if I am arguing something contrary to that point.
...
As above, you are right, my description is inadequate. It helps noone when I leave room for misinterpretation.
X > all('describes' member of z)
Do these deductions follow from the premise?
I think I've clarified and corrected the various points you raised with regard to this.
I am optimistic that if I can communicate this, it will go a long way to A) communicating a thought, B) showing why I'm coming to the conclusions that I am.
...
You say that you picked one piece out of the maelstrom that you could connect with. Good. If we can reach agreement on just one thing it gives us a foundation to work on  which would be nice.
You suggest that we could extend our situation such that in addition to
(perfectly'describable) vs (imperfectly'describable')
we also have
(imperfectly'describable') vs not(imperfectly'describable') where not(imperfectly'describable') means (not'describable'toanydegree).
This would then set up three distinct groups, rather than two: (perfectly'describable'), (imperfectly'describable') and (not'describable')
Generally, if it is possible to logically extend a binary system to a trinary system (and presumably beyond) then reasoning based only on properties of a binary system no longer apply.
The problem with the argument as I've presented it above is that we've already shown that (imperfectly'describable') and (not'describable') are different labels for the same thing. A thing cannot be its own notthing. From the perspective of (perfectly'describable'), (imperfectly'describable') and (not'describable') are exact synonyms.
It looks to me like casual English use of adjectives: is, not, imperfectly, partially, mostly, slightly, ... ; vs mathematical adjectives: is and isnot; could cause problems.
As much as mathematics does build structures far more complicated than boolean switches, everything must resolve down to either 'x = y' or 'x != y' for all applicable x and y. x being both equal to y and not equal to y leads to a "segment fault error  please reboot  press 'okay' to disappear down a logic hole". Likewise x cannot be neither '= y' nor '!= y'.
'x is partially equal to y' isn't even an option that is on the table.
Confusing the English concept of description for the axiomatic mathematics concept of 'description' would probably lead to a great deal of confusion all around. In English, a 'partial description' is a completely mundane idea. Yet, where all propositions are excluded middle: 'partial description' = 'imperfect description' = 'notadescription' = 'everythingthatisnota'perfectdescription''.
...
In bringing up paradoxes you appear not only to be able to understand me but also to predict me. I am genuinely impressed given that I do understand the points you've made and the extent to which I was not being nearly as clear as I wanted to be.
Yes, I think it is evident that the thing that I may or may not have described yet, is, at very least, close to the paradoxical side of the line.
I also think that the above is a slight generalisation of axiomatic mathematics (maybe that will become clearer once I've actually conveyed all the information). But before that, I think that me successfully communicating any set of premises and logical deductions would make everyone feel more comfortable.
...
As you rightly observe, the concept of the excluded middle is important to me. As such, I'm sat over here wondering why people are poopooing such an important and fundamental choice in how to approach formal knowledge.
However, given my use of the phrase is making things worse, I'll endeavour to avoid it. The essential parts for the moment are "all propositions are either x or !x", and the binary nature of x and !x such that there is no semix.
Much of what you are saying I agree with. For example, Your careful analysis of my attempt to communicate a set of premises and logical deductions clearly explained where I was being ambiguous or straight up unclear. I hope my followup attempt fixes these issues.
There are points where I don't fully agree with you  but even here you have explained your position, shown how it differs from mine and presented a good argument to justify that view. I will, of course, argue these points (ideally, 'argue': to discuss until consensus is reached; as opposed to blindly championing a cause). I see that your points are well founded and that compelling evidence needs to be shown if I am arguing something contrary to that point.
...
As above, you are right, my description is inadequate. It helps noone when I leave room for misinterpretation.
 A system X is specified such that, in principle, all well formed propositions within and relating to X resolve to either true or false. Any conceivable alternative state(s) than 'true and false' are explicitly rejected (rename the labels  but there is no semitrue). As you note, being explicit that all propositions are either true or false should be redundant.
 z is a set of every instance of a thing.
 thing is just a thing who's definition is to be consistent with the other stated assumptions.
 'describe' is a set of actions or states that can be applied to all members of z.
 X is able to construct every possible instance of ('describes' member of z)
 X is 'describable'.
X > all('describes' member of z)
 X must be a member of z.
 All 'describe' are members of z.
 All z are 'described' within X.
 notz cannot be 'described' within X.
 This is a closed loop (Nothing from outside X can impinge on anything inside X).
 Every 'description' requires a 'description' in order to be 'described'.
 All chains of 'describing' 'descriptions' must be circular.
Do these deductions follow from the premise?
I think I've clarified and corrected the various points you raised with regard to this.
I am optimistic that if I can communicate this, it will go a long way to A) communicating a thought, B) showing why I'm coming to the conclusions that I am.
...
You say that you picked one piece out of the maelstrom that you could connect with. Good. If we can reach agreement on just one thing it gives us a foundation to work on  which would be nice.
You suggest that we could extend our situation such that in addition to
(perfectly'describable) vs (imperfectly'describable')
we also have
(imperfectly'describable') vs not(imperfectly'describable') where not(imperfectly'describable') means (not'describable'toanydegree).
This would then set up three distinct groups, rather than two: (perfectly'describable'), (imperfectly'describable') and (not'describable')
Generally, if it is possible to logically extend a binary system to a trinary system (and presumably beyond) then reasoning based only on properties of a binary system no longer apply.
The problem with the argument as I've presented it above is that we've already shown that (imperfectly'describable') and (not'describable') are different labels for the same thing. A thing cannot be its own notthing. From the perspective of (perfectly'describable'), (imperfectly'describable') and (not'describable') are exact synonyms.
It looks to me like casual English use of adjectives: is, not, imperfectly, partially, mostly, slightly, ... ; vs mathematical adjectives: is and isnot; could cause problems.
As much as mathematics does build structures far more complicated than boolean switches, everything must resolve down to either 'x = y' or 'x != y' for all applicable x and y. x being both equal to y and not equal to y leads to a "segment fault error  please reboot  press 'okay' to disappear down a logic hole". Likewise x cannot be neither '= y' nor '!= y'.
'x is partially equal to y' isn't even an option that is on the table.
Confusing the English concept of description for the axiomatic mathematics concept of 'description' would probably lead to a great deal of confusion all around. In English, a 'partial description' is a completely mundane idea. Yet, where all propositions are excluded middle: 'partial description' = 'imperfect description' = 'notadescription' = 'everythingthatisnota'perfectdescription''.
...
In bringing up paradoxes you appear not only to be able to understand me but also to predict me. I am genuinely impressed given that I do understand the points you've made and the extent to which I was not being nearly as clear as I wanted to be.
Yes, I think it is evident that the thing that I may or may not have described yet, is, at very least, close to the paradoxical side of the line.
I also think that the above is a slight generalisation of axiomatic mathematics (maybe that will become clearer once I've actually conveyed all the information). But before that, I think that me successfully communicating any set of premises and logical deductions would make everyone feel more comfortable.
...
As you rightly observe, the concept of the excluded middle is important to me. As such, I'm sat over here wondering why people are poopooing such an important and fundamental choice in how to approach formal knowledge.
However, given my use of the phrase is making things worse, I'll endeavour to avoid it. The essential parts for the moment are "all propositions are either x or !x", and the binary nature of x and !x such that there is no semix.
 doogly
 Dr. The Juggernaut of Touching Himself
 Posts: 5493
 Joined: Mon Oct 23, 2006 2:31 am UTC
 Location: Lexington, MA
 Contact:
Re: Misunderstanding basic math concepts, help please?
Replacing things with x and !x doesn't help though if you actually have the associated baggage still in your head or lurking in a shadow somewhere. True, Provable, and Computable (and their ! friends) are the same insofar as they are just binary tokens, but, they are not just binary tokens. "Describable" may more or less map onto some of these known concepts? It's hard to say.
It would be best to completely abandon the baggage you're bringing and work through the exercises with pure tokens, and then ramp things back up.
It would be best to completely abandon the baggage you're bringing and work through the exercises with pure tokens, and then ramp things back up.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

 Posts: 450
 Joined: Wed Sep 21, 2011 3:44 am UTC
Re: Misunderstanding basic math concepts, help please?
I'll take #12 from the worksheet.
Then why the objections? The whole can of worms about imperfect descriptions started when we were talking about explaining formal logic to a human being who doesn't already understand it. It's at that level that we're stuck with informal languages like English, and stuck with imperfect, partial descriptions. But once we take it for granted that we all understand formal logic, *then* it might make sense to expect perfect descriptions under that assumption of a shared formal language. But whether that expectation is sensible or not still depends on exactly what you mean by a "perfect description".
Looking at your attempted explanation of what you mean by a "perfect description" (the stuff about a system X and set of z's and so on), it's still pretty vague and has some undefined terms, (what does it mean for something to "impinge on" X, for example), but it looks like you're setting up a selfreference paradox. (X is a member of z, X describes all members of z, therefore X describes X). This is why I brought up that we start with some informal language like English, because then you describe X using English, rather than describing X using X, and the selfreference goes away.
Spoiler:
Treatid wrote:In English, a 'partial description' is a completely mundane idea.
Then why the objections? The whole can of worms about imperfect descriptions started when we were talking about explaining formal logic to a human being who doesn't already understand it. It's at that level that we're stuck with informal languages like English, and stuck with imperfect, partial descriptions. But once we take it for granted that we all understand formal logic, *then* it might make sense to expect perfect descriptions under that assumption of a shared formal language. But whether that expectation is sensible or not still depends on exactly what you mean by a "perfect description".
Looking at your attempted explanation of what you mean by a "perfect description" (the stuff about a system X and set of z's and so on), it's still pretty vague and has some undefined terms, (what does it mean for something to "impinge on" X, for example), but it looks like you're setting up a selfreference paradox. (X is a member of z, X describes all members of z, therefore X describes X). This is why I brought up that we start with some informal language like English, because then you describe X using English, rather than describing X using X, and the selfreference goes away.
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:
 X must be a member of z.
 All 'describe' are members of z.
 All z are 'described' within X.
 notz cannot be 'described' within X.
 This is a closed loop (Nothing from outside X can impinge on anything inside X).
 Every 'description' requires a 'description' in order to be 'described'.
 All chains of 'describing' 'descriptions' must be circular.
Do these deductions follow from the premise?
I don't follow this.
Let's break this down even simpler.
"X must be a member of z." z is a set that contains some elements. One element of z is X. Good.
"All 'describe' are members of z." All elements of z have property Y. Good.
"All z are 'described' within X" This is nonsense. You've completely reversed the associations from the two previous lines. What follows is not that z has property Y, but that X does.
All of the lines that follow pretty much suffer from the same problem... you've changed the association from X being the element and z being the set to z being the element and X being the set.
Re: Misunderstanding basic math concepts, help please?
Why the aversion to the worksheet, Treatid? Everyone in the thread has been asking you to work on it so that we can understand you better.
Here's even better questions:
1) Why do YOU think it is so important to all of us in the thread that you can solve the problems on the worksheet?
2) Why do you think it is OK (in light of the fact that we all think it is so important) for you to continue to ignore our requests for you to work on it?
Here's even better questions:
1) Why do YOU think it is so important to all of us in the thread that you can solve the problems on the worksheet?
2) Why do you think it is OK (in light of the fact that we all think it is so important) for you to continue to ignore our requests for you to work on it?
Re: Misunderstanding basic math concepts, help please?
Treatid wrote:
 A system X is specified such that, in principle, all well formed propositions within and relating to X resolve to either true or false. Any conceivable alternative state(s) than 'true and false' are explicitly rejected (rename the labels  but there is no semitrue). As you note, being explicit that all propositions are either true or false should be redundant.
 z is a set of every instance of a thing.
 thing is just a thing who's definition is to be consistent with the other stated assumptions.
 'describe' is a set of actions or states that can be applied to all members of z.
 X is able to construct every possible instance of ('describes' member of z)
 X is 'describable'.
So let me try to summarize this in my own words, to see if I've got it.
We've got a set z, which is just a collection of some objects. (What type of objects they are is not particularly relevant.) We've got a collection of functions (treating descriptive sentences as unary functions that return 'true' or 'false') that are defined on precisely z, and this collection is called 'describe'. Now, z is "every instance" of the describable things, that is, things within the domains of the 'describe' functions. X is a system that takes the Law of the Excluded Middle as an axiom, and it also has among its wellformed formulas all statements about how the functions in 'describe' behave or could behave upon the elements of z. Furthermore, X itself is describable, so the functions of 'describe' can be applied to X.
Right off the bat, I don't see why such an X should exist. Putting the Law of the Excluded Middle aside (because it doesn't seem particularly relevant) I see no reason that we should be able to find among the elements of z a system that contains wellformed formulas about all the members of z, including itself. Especially with 'describe' defined as nebulously as it is, I would need to see an argument as to why such an X actually exists.
I get that you want to leave 'describe' defined nebulously so that you can harness the natural language properties of the word 'describe' (or at least, that's what it seems like you're doing to me). I'm not convinced that it's possible to get all the naturallanguage properties you want into a strict mathematical definition. Things like Russell's Paradox and Godel's Incompleteness Theorem struck down notions like "a system where I can write a description and have a set of all things that match that description" and "a system where I can prove all the true things about arithmetic" as impossible. (Obviously, I'm paraphrasing in these two examples; the problems are more subtle.) Why do you think your notion *is* possible?
But putting that aside, let's say that such an X does exist, so that I can evaluate the rest of your argument.
X > all('describes' member of z)
This notation is really weird.
 X must be a member of z.
 All 'describe' are members of z.
 All z are 'described' within X.
 notz cannot be 'described' within X.
 This is a closed loop (Nothing from outside X can impinge on anything inside X).
 Every 'description' requires a 'description' in order to be 'described'.
 All chains of 'describing' 'descriptions' must be circular.
1) Okay.
2) I don't know exactly what you mean by "all 'describe' ". You said that " 'describe' is a set of actions or states"; do you mean that those functions in describe are themselves in z? Or are you saying that everything which is describable (in the sense which I've defined above using bold) is in z? I'd agree with you under the latter interpretation, but under the former interpretation I don't see why that should hold.
3) Please define 'described'. 'Describe' was just a collection of functions, but now you seem to want to use the semantic properties of the English word "describe". This is dangerous territory you're treading, which is to say that it's very easy to make a logical error when you're not careful with the meanings of your terms.
4) For whatever meaning of 'described' you end up with, it's probably safe to say that things that are not in z cannot be 'described' (assuming, of course, that notz is the collection of things not in z), since they're not in the domain of the 'describe' functions, by definition of z. I'll point out that this conclusion is not related in any way to X having the Law of the Excluded Middle as an axiom, because *we're* not reasoning using the axioms of X.
5) I don't see how you get to this at all. Or rather, I'm not sure exactly what you mean by "closed loop", and "impinge". Just because X models the actions of the functions of 'describe' on the elements of z, that doesn't mean that nothing else can model those actions, so I'm not sure what you're driving at.
6) Please define 'description'. Your use of this term here makes me think that you've conflated the English language meaning of "describe" with the one you set out at the beginning of this argument, which was that 'describe' was a a "set of actions or states". Additionally, your conclusion here seems to indicate that the correct interpretation for (2) was that all functions of 'describe' are in 'z', which I don't think you've demonstrated.
7) Please define "chain" and 'describing'. Also, why do these "chains" have to be circular? I'm pretty sure you're trying to argue that since you can go back forever in these "chains", they have to loop back on themselves, but they could also just go back forever without looping, the way the negative integers go back forever without looping back to 1 ever.
Without definitions of several terms in your argument, I can't evaluate it fully. The problems I point out are only the ones that seem to apply no matter what definitions you end up supplying.
You suggest that we could extend our situation such that in addition to
(perfectly'describable) vs (imperfectly'describable')
we also have
(imperfectly'describable') vs not(imperfectly'describable') where not(imperfectly'describable') means (not'describable'toanydegree).
What I suggest is that you're careful with the meanings of your terms. Like I said before, when I hear imperfectly'describable', I do not think of that as including all things that are not perfectly'describable'. Rather, I think of it as including all 'describable' things which are not perfectly'describable', and I do not inherently think of all 'descriptions' as perfect.
Also, I would never say that not(imperfectly'describable') is the same as not'describable'toanydegree.
This would then set up three distinct groups, rather than two: (perfectly'describable'), (imperfectly'describable') and (not'describable')
Yeah, sure. You're attaching a lot more significance to the existence of three groups than I would.
Generally, if it is possible to logically extend a binary system to a trinary system (and presumably beyond) then reasoning based only on properties of a binary system no longer apply.
What it means is that your two groups didn't encapsulate everything. It happens.
The problem with the argument as I've presented it above is that we've already shown that (imperfectly'describable') and (not'describable') are different labels for the same thing. A thing cannot be its own notthing. From the perspective of (perfectly'describable'), (imperfectly'describable') and (not'describable') are exact synonyms.
...what? I don't even know what the perspective of an adjective is. Again, I'm taking my best guess as to what you're saying here, but:
It sounds like you've confused a conditional statement with its converse. It is certainly the case that
If something is imperfectly'describable', then it is not perfectly'describable'.
However, it does not follow from that that
If something is not perfectly'describable', then it is imperfectly'describable'.
Similarly, I don't see how you could conclude that
If something is not perfectly'describable', then it is not'describable'.
Or at least, I don't see how you could conclude that unless you have some interesting definitions for perfectly'describable, imperfectly'describable', and 'describable'. I'm starting to get a little worried here that we're just talking about two different things, and it would assuage me if you could supply your definitions of, or at least some exposition on, the three terms I mentioned earlier in this paragraph.
It looks to me like casual English use of adjectives: is, not, imperfectly, partially, mostly, slightly, ... ; vs mathematical adjectives: is and isnot; could cause problems.
Hence my request for your definitions/expositions.
As much as mathematics does build structures far more complicated than boolean switches, everything must resolve down to either 'x = y' or 'x != y' for all applicable x and y. x being both equal to y and not equal to y leads to a "segment fault error  please reboot  press 'okay' to disappear down a logic hole". Likewise x cannot be neither '= y' nor '!= y'.
'x is partially equal to y' isn't even an option that is on the table.
If we have three things, x, y, and z, with x != y, x != z, and y != z, then what does notx represent? Certainly, y is not x, but I would balk at the idea of your saying that y is notx. After all, z is not x, but it can't be the case that both y is notx and z is notx, since y != z.
Unless you mean by "y is notx" that "y has the property 'notx' ", but in that case, it would be improper to assert that y is the same thing as notx, since y is a thing, and notx is a property.
Rather than any "equal or not equal, there is no partial equality" issue, I think that what I said in the above paragraphs is the heart of the problem I have with your describability rhetoric. You're abusing the word "is" somehow, conflating two of its meanings as demonstrated in the following two sentences: "Barack Obama is the President of the United States", and "The sky is blue". One of these equates two things, and one says that a thing has a property.
Confusing the English concept of description for the axiomatic mathematics concept of 'description' would probably lead to a great deal of confusion all around. In English, a 'partial description' is a completely mundane idea. Yet, where all propositions are excluded middle: 'partial description' = 'imperfect description' = 'notadescription' = 'everythingthatisnota'perfectdescription''.
What is a "perfect description", then? You want to leave the English language definition behind, so what's your proposed alternative? Because it seems to me like you're making a lot of deductions based on the English language meaning and properties of "perfect description" (For instance, you conjugated the term 'describe' as though it were an English verb earlier).
Regarding your equation at the end: saying that an imperfect description is not a description does not follow in any way, shape, or form from the Law of the Excluded Middle. It follows from your asserting that imperfect descriptions are not descriptions. If you want to do this, fine, but DO IT.
If you think it does follow from the Law of the Excluded Middle, then prove it. Start from the actual Law of the Excluded Middle, which, for the thousand and third time (someone else called you on it since the last time I did), deals with truth values. Derive that the only descriptions are perfect descriptions. Justify your steps, and do not use that the only descriptions are perfect descriptions as a justification. I know, this might be hard, since logical proofs are tricky. If you're feeling unsure, might I suggest some practice problems?
As you rightly observe, the concept of the excluded middle is important to me. As such, I'm sat over here wondering why people are poopooing such an important and fundamental choice in how to approach formal knowledge.
However, given my use of the phrase is making things worse, I'll endeavour to avoid it. The essential parts for the moment are "all propositions are either x or !x", and the binary nature of x and !x such that there is no semix.
That doesn't really help. The problem isn't the phrase, it's that you're invoking the law in places where, to us, it doesn't say anything at all of note, so that when you draw incredibly strong conclusions from it it seems highly suspect. What would help a whole, whole lot is if, every time you wanted to invoke the Law of the Excluded Middle, you did so by mentioning a proposition and its negation, and claiming that
one of the two is true. Then, demonstrate the deductions that lead from there to the conclusion you actually want to draw. Currently, it seems to me (and I imagine others in this thread) that your arguments go "1) 'Law of the Excluded Middle!' 2) *some magic happens behind the scenes* 3) The conclusion you want to draw".
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Re: Misunderstanding basic math concepts, help please?
Twistar wrote:Why do YOU think it is so important to all of us in the thread that you can solve the problems on the worksheet?
You know, that's a good question.
Why is it so important to you? Can't you have a meaningful discussion about logic and the structure of axiomatic mathematics, without forcing people to do textbook problems?
I do agree that we need to find common grounds for this discussion to be useful, but demanding the other person to "speak as we speak, or be ignored" is hardly an effective way to establish common grounds.
Also, remember that different people process information in different ways. Some (like you and me and most of the other people writing on this thread) get things more clearly when they are written in formal mathematical code, while others (like Treatid here and  actually  the majority of the world's population) just get confused by such definitions.
Now, I understand that some formality is inescapable when discussing a topic such as this. But it should really be kept to the possible minimum if you are seriously interested in actual communication with nonmathematicians. And there's no way that demanding a person to do worksheet problems as a prerequirement for conversation, can be regarded as anything other than a dick move. Sorry, but that's the way I see it.
So let's try to get back on track:
Treatid, can you please tell me what's the actual topic you wished to discuss in this thread? I am assuming that all the formal talk, not to mention that absurd "worksheet demands" thing, is anything but relevant to your actual questions...
 gmalivuk
 GNU Terry Pratchett
 Posts: 26577
 Joined: Wed Feb 28, 2007 6:02 pm UTC
 Location: Here and There
 Contact:
Re: Misunderstanding basic math concepts, help please?
Overruled.
The request that Treatid do some simple worksheetstyle formalism, or at least explain the unwillingness to do so, is a reasonable one, especially since we've all seen how frequently relying on vague intuitions about natural language has led these discussions astray.
Sure, the majority of the population gets confused by formal mathematical definitions, but the majority of the world's population doesn't regularly start threads in the skcd math forum about some perceived deficiency in those definitions.
If Treatid would like to join them in their ignorance of formal language, then Treatid is also more than welcome to join them in their absence from this forum.
The request that Treatid do some simple worksheetstyle formalism, or at least explain the unwillingness to do so, is a reasonable one, especially since we've all seen how frequently relying on vague intuitions about natural language has led these discussions astray.
Sure, the majority of the population gets confused by formal mathematical definitions, but the majority of the world's population doesn't regularly start threads in the skcd math forum about some perceived deficiency in those definitions.
If Treatid would like to join them in their ignorance of formal language, then Treatid is also more than welcome to join them in their absence from this forum.
Re: Misunderstanding basic math concepts, help please?
gmalivuk wrote:The request that Treatid do some simple worksheetstyle formalism, or at least explain the unwillingness to do so, is a reasonable one, especially since we've all seen how frequently relying on vague intuitions about natural language has led these discussions astray.
I agree that demanding Treatid to explain himself more clearly is not only reasonable, but a requirement to have a fruitful discussion. This thread is, in itself, an excellent example of why this is necessary.
But there's more than one way to do this, and quite clearly Twistar's approach isn't working. And even though I'm a person who is wellversed in mathematics and logic, I don't find it at all surprising that Treatid is refusing to cooperate with Twistar's requests. I myself find these demands to be arrogant and borderline offensive.
I also don't understand what you mean by "overruled". Are you simply voicing the official xkcd position, or prohibiting me from voicing my opinion on this? If it is the latter, I will comply.
At any rate, in either case, my own request to Treatid still stands:
Treatid, can you please restate your original question? I suggest we go through it stepbystep, and not proceed to the next step until we both agree that we're on the same page. Okay?
Who is online
Users browsing this forum: No registered users and 10 guests