Cauchy wrote:It's unclear what exactly "absolute" and "relative" mean here, and your stance conveniently seems to relieve you of the need to have hard definitions.
Let me turn that around on you.
Mathematicians cannot define what absolute or relative means; they cannot define dimensions or integers; they can't define what axiomatic mathematics is; they can't define what true means. In short, mathematicians have never, ever defined anything - not even a little bit.
Whoops. Did I miss out "in an absolute sense"? I surely missed that they can't even define what 'define' means.
Yes, how convenient it is for me that mathematicians have not ever defined anything in an absolute sense.
If you have an actual argument - bring it. Meanwhile, you can shut me down by producing a single absolute definition. Perhaps you could try your hand at defining axiomatic mathematics?
But that is the problem, isn't it. You've conceded that mathematics does not define anything in an absolute sense... But you still think that mathematics does define things.
"Yeah - maybe we don't have absolute definitions - but we have good enough definitions. Technically they aren't absolute truths; but we still know what we mean when we say that something is true."
No. You don't know anything. I don't know anything either in this sense. My entire thesis at this point is that it is impossible to know anything. There is no fault in not being able to define 'true' within an axiomatic system. I'm not saying axiomatic mathematics is wrong because it can't formally define 'true' within a given axiomatic system. I'm saying axiomatic mathematics is wrong for trying to define anything. Mathematicians are wrong for pretending that they have defined anything, to any degree.
The one success of axiomatic mathematics is that it has demonstrated beyond reasonable doubt that the concept of defining something is null and void.
And your response?
"Yeah - well technically yes. But we totally know what axiomatic mathematics is. We totally know what the axiom of choice is. We totally know what dimensions are. We just can't quite write that knowledge down on paper."
There is no middle road here. Do you know something? No? Then you don't know anything.
Yes, of course we have a concept of knowledge. We have libraries full of books. I know when my mother's birthday is. I know how to switch on a computer, select a forum and type characters into the little window that appears.
And then I type the characters: "It is impossible to know anything" and your brain goes "squeak" and you get an intense urge to reach through the screen and throttle some sense into me.
But it is exactly this dichotomy that is the problem. We have a concept of knowing that appears rational and justifiable. You know what it means to say that you know what the theory of evolution is. You know what knowledge is. You know what it means to define something. These are perfectly normal words used in a productive manner all the time. It seems utterly perverse to tell an English speaking person that they don't know what the words they are using mean.
And then we have axiomatic mathematics. An attempt to properly formalise those meanings into a single consistent framework so that there is no messy informal doubt; so that all those arguments over the precise meaning of a given word can be settled in a formal environment with none of that tedious messing around with subjective interpretations.
And what happens?
You can't define true. You can't define consistent. You most definitely can't demonstrate consistency. You can't even define 'define'.
There is not a single word, phrase, group of symbols or anything else that you can assign a single definite meaning to.
And it isn't that you just haven't worked out how to do it yet. You have formal statements saying that there are fundamental limits on what axiomatic mathematics can do.
Whatever it is that axiomatic mathematics is trying to do (which you can't define), it isn't doing it. But because you can't define what it is that you aren't doing you are able persist in the fiction that you are doing something despite all the results telling you that you really, really are not defining anything in any sense.Gargh! Again!
Right now you (as a group) are trying to tell me that it doesn't matter that axiomatic mathematics can't define anything and that it has always been a relativistic theory. You are arguing that it was never intended to define anything. Actually - I'm pretty sure you don't know what you are arguing - except you know you are arguing against whatever it is you think I'm arguing.
Yahweh! you are so full of crap.
You're like a kitten trying to cover up its shit on a wood floor.
Me: You can't define anything in an absolute sense.
You: Yeah. I suppose technically. But we were never trying to define things in an absolute sense. Yeah; that's it. We are defining things in a sense that is totally not absolute.
Me: Really? what other meaning/sense of definition are you using?
You: Umm... Well... we know it when we see it...
Me: But...? You can't actually define the sense of definition that you think you might be using, can you? You think you know what you are doing; but you can't quite put into words exactly what that might be. You're sure that axiomatic mathematics works in the way that you understand it; you're sure that you are being consistent in your behaviour - but you can't say what axiomatic mathematics is. You can't say what consistent is. You can't say what true is.
The first mantra of mathematics is: Define your terms.
And here you are trying to tell me that axiomatic mathematics has always been relative and it doesn't matter that you haven't defined anything. At all. Ever.Lie to yourself all you like
You can tell me that I'm misunderstanding axiomatic mathematics and the Laws of Thought all the live day long. We could spend years arguing back and forth.
And the reason we could spend years arguing over exactly what is, and is not, the correct understanding of axiomatic mathematics and the Laws of Thought is that they aren't defined. At all. Not even a little bit.
This is the very problem that formal mathematics was intended to solve. If you don't define your terms then you can twist and turn and change your meaning part way through in order to argue any position you want to.
The trouble is that the result of this attempt to formalise definitions is to demonstrate that it is impossible to formalise definitions.
So here we are with you inventing new ways to interpret what axiomatic mathematics is and what it is trying to do.
And yes - you have a point - your interpretation is as valid as mine. Which is to say - neither of us can prove that axiomatic mathematics or the laws of thought have a single definite meaning exclusive of all other meanings.
The difference between you and me is that you are in denial. You think it is possible to know things. You think that there is a possibility of a fixed reference frame. You think you know what (x = x) means.Oh - and incidentally
In order to determine if (x = y) you need to know what x and y are. You need to define x and y (and equality). You need to assume that x and y can have known values. You are assuming that x and y are absolutes.
If you are going to dick around re-interpreting fundamental assumptions of mathematics it might be a good idea to make sure that your re-interpretation is actually making the point you intended rather than... you know... the exact opposite point.
Whether you take the First Law as an argument about x or an argument about equality; it is clearly a statement of absolutes. That there is something that can be known. Whether that something is x, equality or identity is missing the point that it is a statement of certainty - of absolute knowledge - a thing is a thingDon't be confused by the fact that the English Language works (to some degree)
You know what knowledge is.
Except that axiomatic mathematics has demonstrated that is is impossible to know anything in an absolute sense.
There is a mismatch here. A false assumption is being made.
Axiomatic mathematics has been crystal clear in telling us that it isn't possible to define anything.
Yet our personal experience of knowledge compels us to believe in the existence of definitions despite all the many ways in which formal analysis of axiomatic mathematics shouts, screams and begs us to understand that it isn't possible to define anything.
Thousands of years of Euclidean geometry: How could anyone say that Euclidean Geometry hasn't been defined? As mathematicians you've been working with Euclidean Geometry since you first learned to count (even if nobody named it the number line when you first learned 1 + 1).
You are sure that you've defined stuff. Even if you concede that the definition isn't absolute and universal - you just know that you have defined stuff.
So when I tell you that nothing has ever been defined to any degree - that it is impossible to define anything in any way then of course you are going to rebel. You feel right down to your bones that you know what Euclidean Geometry is.
In your mind, 'definition' and 'existence' are near synonyms. Denying the existence of definitions feels like denying the existence of the universe - or our own thoughts. You don't have to prove your own existence to yourself. You shouldn't have to justify the existence of definitions - they just are - it is what humans do - we define the world around us. It is what makes us humans. If we have never defined anything then (error... memory segment fault... does not compute... We have always been at war with East asia... axiomatic mathematics was always meant to be relativistic... non absolute definitions are definitely a thing (that I just can't quite define at this precise moment)... Fizz... Pop... Whirr...)Specifically
The standard reaction here when conceding that there are no absolute definitions has been to immediately imply the existence of satisfactory non-absolute definitions.
"Obvious really. Makes sense. We know we have English language definitions - so if we don't have absolute definitions then in the very worst case we have English language definitions. And axiomatic mathematics is more formal than English language so axiomatic mathematics definitions are automatically a step up from English language definitions - even if they don't reach all the way up to the lofty goal of absolute definitions."
Yes... That is all very plausible. Utter bollocks - but plausible bollocks.
You are getting away with sloppy thinking because you don't have any definitions to constrain yourself.
You can't even define 'definition'. What makes you think you are defining anything? No, really! Why do you think you've defined ZFC?
Yes, English works. Yes, English contains the word 'definition'.
And still I can sit here and confidently type: There has never been a definition of anything. Ever.
We can argue and fuss over what I mean by definition and what you mean by definition. Eventually we will agree that there is no definite, absolute, epistemically certain way to define definition.
"Oh - but there is totally this other meaning of definition."
Fine: tell me what it is so that I am certain that we agree on the definition.
"But that would require an absolute definition."
Yes. Quite. That is the point.Reminder
I'm not being nihilistic. I'm not saying we should just give up on everything and live in caves.
I'm saying that everything we have achieved has been despite never defining everything. We don't need to define things. Stop trying. Stop pretending that anything is in any way a stand-in for an absolute definition (i.e. stop pretending that you have defined axiomatic mathematics, ZFC, dimensions, true, consistent, or anything else).
It isn't that the concepts just happen not to have been defined. It is that they are undefinable. It is not possible to define anything. To attempt to do so is a category error. There cannot be a fixed point. There cannot be a thing that only has one single meaning; one single definition.
Axiomatic mathematics cannot exist. No single definition can be un-entangled from the infinitely many alternatives. There is no possible mechanism to choose one meaning over every other meaning.
“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”
The malleability of words has been known, probably since mankind first grunted. "Was that a grunt yes? or a grunt no?"
Yes is it a pain when trying to communicate and the other person is mis-understanding your words. Tough. There aren't any shortcuts. There is no way to magically transfer your interpretation of a word to someone else. Symbols are just symbols. There is no special set of symbols that suddenly everyone will realise has a fixed definite meaning that can then be extrapolated to other symbols. They are just symbols.Polite
I want to be polite. By being aggressive in my posts I get your backs up and it is that much harder for you to read anything I write in a sympathetic light. It makes you grasp at the flimsiest of counters just so you can stick it to me.
However, you are currently trying to tell me that axiomatic mathematics has always been a purely relativistic system, the Laws of Thought don't assert the assumption of absolute values and it was never part of axiomatic mathematics to define anything. (oh yes - and in the absence of being able to define axiomatic mathematics you still somehow know what axiomatic mathematics is)
So I'm trying another tack: criticise your ability to think rationally. Not a tactic that usually works on religious conviction, but maybe a couple of you take pride in your ability to think rationally and will notice that if you haven't defined anything, then you haven't defined axiomatic mathematics.