I already did above. The axioms of some given set theory are just rules that you must follow when using that set theory. Different set theories have different axioms and so different rules. Given that there's no connection between using some set theory and believing in the mind-independent existence of abstract mathematical objects, there's no hypocrisy in using some set theory and being a mathematical antirealist. — Michael

"Mathematical antirealist" is also a "rule". It states an ontological principle, or rule. It is a rejection of mathematical objects. The rules of set theory are inconsistent with this rejection of mathematical objects, because set theory assumes mathematical objects, as a foundational premise. Therefore you must assume mathematical objects, as a fundamental premise, to be able to follow the rules of set theory. This activity is contrary to the ontological belief stated as "mathematical antirealist", and is therefore hypocrisy for anyone claiming to be a mathematical antirealist.

Your position is like arguing that it's hypocritical to play chess if I do not believe that the rules of chess correspond to some mind-independent fact about the world. — Michael

I don't see the relevance. You do not need to accept the premise of "mathematical objects" to play chess. You do need to accept the premise of "mathematical objects" to follow the rules of set theory.

The key point here, is that imagination does not require sensation of whatever it is that is imagined. — Metaphysician Undercover

I imagine a unicorn by picturing a unicorn.

How do you imagine a unicorn if you don't picture a unicorn?
===============================================================================

The point though, is that in the case where you used "=" to signify identity, it is not a mathematical usage. — Metaphysician Undercover

"1 = 1" is a mathematical expression. The expressions "Twain = Clemens" and "sugar = bad" are not mathematical expressions.

Similarly, the word "infinity" has one meaning in a formal set theory and a different meaning in everyday natural language

From Frege's "Context Principle", the meaning of "=" and "infinity" depend on their contexts.

I don't see the relevance. You do not need to accept the premise of "mathematical objects" to play chess. You do need to accept the premise of "mathematical objects" to follow the rules of set theory. — Metaphysician Undercover

You need to accept the premise of queens and kings and pawns to play chess, but accepting this premise doesn't commit you to "chess realism".

You need to accept the premise of a murderer and a victim when playing Cluedo, but accepting this premise doesn't commit you to "Cluedo realism".

And so accepting the premise of mathematical objects when using set theory doesn't commit you to mathematical realism.

When using set theory, mathematical objects "exist" only in the sense that queens "exist" in chess and a murderer "exists" in Cluedo, i.e. not in any realist sense.

Set theory begins with the assumption of mathematical objects, hence it is based in Platonic realism. — Metaphysician Undercover

You don't need to believe in Platonic realism to use set theory. Its axioms are just rules to follow when "doing" maths. — Michael

I agree. I didn't say you need to believe in the truth of the principles you employ. However, it's hypocrisy to say "I'm a mathematical antirealist" and then go ahead and use set theory. But that sort of hypocrisy is extremely commonplace in our world, it's actually become the norm now. Very few people make the effort to understand the metaphysics which they claim to believe in, and whether it is consistent with the metaphysics which supports the theories which they employ in practise. — Metaphysician Undercover

Is it hypocritical for a mathematical formalist to use set theory? I think the differences between the philosophical schools in mathematics don't actually matter so much because the differences are in the realm of metaphysics. If for a Platonist the abstract mathematical objects exist and for the formalist it's just basically something compared to an eloquent game, what's the actual difference?

The way I see it the difference between anti-realists and realists (Platonists of some sort?) is things like if mathematical truths are discovered or invented. It doesn't change the math!

The math in set theory is mainly about injections, surjections and a bijection, which we mark usually with "=".

Similarly, the word "infinity" has one meaning in a formal set theory and a different meaning in everyday natural language — RussellA

Problem with Set Theory is that their concept "infinite" means "finite". It breaks the most fundamental principle of Truth.

I imagine a unicorn by picturing a unicorn. — RussellA

Similar, but I wouldn't call it "picturing". Anyway, the point is that this "picturing" does not require a "concrete instantiation", which I assume implies a physical object being sensed.

"1 = 1" is a mathematical expression. — RussellA

When you say "=" signifies identity, "1=1" is not a mathematical expression. Think about it. If "1=1" means that the quantitative value signified by the first "1" is equivalent to the quantitative value signified by the second, then this is a mathematical expression. And in the case of ordinals, if "1" signifies "first", and the expression means that the first is equivalent with the first, as first, then this is also a mathematical expression. But if "1=1" is meant to signify that the thing identified by "1" on the right side is the very same as the thing identified by the "1" on the left side, then it is not a mathematical expression. It is an expression of identity. And, the fact that it is analogous with Twain = Clemens, which is clearly not a mathematical expression is evidence that it is not a mathematical expression.



I could address your examples, but I do not see how they are relevant really. In set theory it is stated that the elements of a set are objects, and "mathematical realism" is concerned with whether or not the things said to be "objects" in set theory are, or are not, objects.

To play chess you must accept the reality of the pieces as objects in order to move them, therefore you must accept "chess reality" to play chess. Since it may not be stated in the rules that the pieces are "objects" the acceptance is only implicit, unlike set theory in which case the rule is explicit, therefore acceptance is explicit.

It seems to me that you do not understand "realism". Do you agree, that to be able to take "an object", manipulate it, move it, do whatever you please with it, or move it according to some set of rules, you need to accept that the object which you are doing this with is "real"? And this implies that believing the things which you are manipulating to be "objects", implies some sort of realism. Or, do you separate "realism" from "objects", so that realism has nothing to do with objects? In which case, what would you base "realism", and consequently "antirealism" in?

To play chess you must accept the reality of the pieces as objects in order to move them, therefore you must accept "chess reality" to play chess. Since it may not be stated in the rules that the pieces are "objects" the acceptance is only implicit, unlike set theory in which case the rule is explicit, therefore acceptance is explicit. — Metaphysician Undercover

You can play chess without a physical board and physical pieces. You can play it with pen and paper if you like; much like we do with maths. Or, if you're very smart, you can play it in your head; again, much like we do with maths.

When playing chess in your head you're not committed to being a realist about the queen you're playing with. When using set theory you're not committed to being a realist about the mathematical objects you're using.

You just follow the rules.

Anyway, the point is that this "picturing" does not require a "concrete instantiation", which I assume implies a physical object being sensed. — Metaphysician Undercover

I can only imagine a unicorn by picturing a unicorn. A picture requires a "concrete instantiation". A "concrete instantiation" can be on a screen or a piece of paper. Both a screen and a piece of paper are physical objects existing in the world. As physical objects in the world, I can sense them.



You can only imagine a unicorn if you know what a unicorn is. How can you know what a unicorn is without having first seen several "concrete instantiations" of it as physical objects in the world?
===============================================================================

But if "1=1" is meant to signify that the thing identified by "1" on the right side is the very same as the thing identified by the "1" on the left side, then it is not a mathematical expression. It is an expression of identity. — Metaphysician Undercover

Identity is a valid part of mathematics.

Wikipedia -Identity (Mathematics)

In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables within a certain range of validity.

I've invented a game. At least I think I invented it. I believe that mathematics is invented rather than discovered, and it is kind of a mathematical game. You can play it with black and white pebbles like you might use for the game Go. It's a solitaire game, though, with no particular aim.

You put the pebbles in rows, from left to right. I'll use B and W to represent the pebbles, but it's nicest to play with natural concrete instantiated objects. There are two rules.

Rule 1. You can make a row by putting two pebbles down like this:

Rule 2. If you have made a row, or some rows, of pebbles, you can join them altogether into one long row, and then put an extra B at the beginning and an extra W at the end.

Let's see some patterns we can make. Using rule 1 we have
We could use rule 1 again.
This is boring. Let's try rule 2. We could make
or
If we took we could make
If we took
we could make

It is possible to interpret these rows of pebbles as multisets. It is possible to interpret some rows as sets. It is possible to interpret some rows as natural numbers. It is possible to interpret the sequence
as counting. It's a pretty cumbersome way of counting. It would be easier to ignore the colours of the pebbles, and just count the pebbles, and interpret the counts as numbers. It is possible to ignore all these interpretations, and just play the game.

For a mathematical antirealist, does any of this constitute hypocrisy?

(@Metaphysician Undercover mostly.)

Time, inclination and patience permitting, I hope to get caught up at some time to responding to the recent various misconceptions, non sequiturs, strawmen, etc. posted in this thread. — TonesInDeepFreeze

What's worse than people trying to do physics without the mathematics?

Apparently, people will also try to do mathematics without the mathematics.

Pointing out their errors simply makes them double down. Sometimes all you can do is laugh and walk away.

Don't have time for all the replies I want to make lately, but this one is easy:

Problem with Set Theory is that their concept "infinite" means "finite" — Corvus

A common definition of 'infinite' in mathematics is 'not finite' You have it completely wrong. Would that you would not persist in posting falsehoods.

we already know you're ugly [...] you appeared highly bothered after I asked if that's why you don't show your face [...] It should be obvious that I'm trolling you. [...] Might loosen up your butthole a little so you can actually poop, my man. — Vaskane

Ugh.

Don't have time for all the replies I want to make lately, but this one is easy:

Problem with Set Theory is that their concept "infinite" means "finite"
— Corvus

A common definition of 'infinite' in mathematics is 'not finite' You have it completely wrong. Would that you would not persist in posting falsehoods. — TonesInDeepFreeze

Me neither. But I try to reply to the posts addressed to me.
Which math textbook says "infinite" means "not finite"?

Apparently, people will also try to do mathematics without the mathematics. — Banno

No one was doing math here. This is philosophy forum, not math. We have been just pointing out that misuse of concepts and definitions, and using them as the premises in their arguments can mislead people with the wrong answers and absurd conclusions.

Pointing out their errors simply makes them double down. Sometimes all you can do is laugh and walk away. — Banno

You claim that you care about philosophy, but don't appear to be doing so. What you seem to be doing here is just codon blindly whoever is on your side whether right or wrong, and laugh and walk away from truths.

yeah, that must be it.

yeah, that must be it. — Banno

:nerd: Be honest to yourself, and try to be your own man. :cool:

I can only imagine a unicorn by picturing a unicorn. A picture requires a "concrete instantiation". A "concrete instantiation" can be on a screen or a piece of paper. Both a screen and a piece of paper are physical objects existing in the world. As physical objects in the world, I can sense them. — RussellA

This is clearly incorrect. We can imagine things without a concrete instantiation. That's how artists create original works, they transfer what has been created by the mind, to the canvas. It is also what happens in dreams, things never before seen are created by the mind.

For a mathematical antirealist, does any of this constitute hypocrisy?

(@Metaphysician Undercover mostly.) — GrahamJ

I can't see the relevance. Your game clearly involves real objects, pebbles, or in the case of your presentation, the letters. Would the antirealist insist that these are not real objects?

Apparently, people will also try to do mathematics without the mathematics. — Banno

Those are the people who say "=" signifies identity in mathematics. They claim to be doing mathematics when they say that "1=1" means that what left 1 signifies is the same as what the right 1 signifies. But that's obviously not mathematics. In mathematics, the left side of the equation always signifies something different from the right side, or else the equation would be useless.

It's one thing for non-mathematicians, who don't know any better, to think that what they are doing is mathematics, when it's not. But it's truly shameful when mathematicians claim to be doing mathematics when what they are doing is not mathematical. As I explained already, that's how they come up with false axioms.

Those are the people who say "=" signifies identity in mathematics. They claim to be doing mathematics when they say that "1=1" means that what left 1 signifies is the same as what the right 1 signifies. But that's obviously not mathematics. In mathematics, the left side of the equation always signifies something different from the right side, or else the equation would be useless.

It's one thing for non-mathematicians, who don't know any better, to think that what they are doing is mathematics, when it's not. But it's truly shameful when mathematicians claim to be doing mathematics when what they are doing is not mathematical. As I explained already, that's how they come up with false axioms. — Metaphysician Undercover

The symbol "=" is defined in ZFC by saying that "A = B" is true if and only if A is B.

They could have used the symbol "#" instead, but they decided on "=".

The symbol "=" is defined in ZFC by saying that "A = B" is true if and only if A is B. — Michael

Yes, and as I've shown over and over again, that definition of "=" is not representative of how "=" is actually used in mathematics. Therefore it is a false definition, designed for some other purpose, foreign to mathematics.

Yes, and as I've shown over and over again, that definition of "=" is not representative of how "=" is actually used in mathematics. Therefore it is a false definition, designed for some other purpose, foreign to mathematics. — Metaphysician Undercover

You're putting the cart before the horse. It's not that we use maths and then retroactively describe what the symbols mean and infer the axioms; it's that we define what the symbols mean, prescribe the axioms, and then use them.

Yes, and as I've shown over and over again, that definition of "=" is not representative of how "=" is actually used in mathematics. Therefore it is a false definition, designed for some other purpose, foreign to mathematics. — Metaphysician Undercover

So when the issue is set theory, isn't then more correct just to talk about a bijection?

Or is that problematic too?

It's not that we use maths and then retroactively describe what the symbols mean and infer the axioms; — Michael

You have this wrong. A study of the history of mathematics will reveal to you that the axioms come about as a representation of usage. We could start with something like "the right angle", and see that the Egyptians were using that concept to create parallel lines and things like that, far before the axiom, the Pythagorean theorem, which represents this usage, was expressed.

As I recently explained in a related thread, since axioms are determined by choice, and used by choice, we must accept that axioms follow usage, they do not determine usage. People can produce whatever axioms they like, but if they are not useful they will not be used, nor become conventional. So, the axioms which become the convention are the ones best representative of what mathematicans are actually doing.

In the case of the axiom of extensionality, it is useful for a purpose other than mathematics. It's use is rhetorical, to persuade people of the usefulness of set theory. It is clearly not true though, because, for example, the order of the elements within a set is not accounted for. So, sets which are said to be identical may have the same elements in a different order. But in any true sense of "identity" order is an essential feature. Therefore the rhetorical use of this axiom is really a matter of deception.

So when the issue is set theory, isn't then more correct just to talk about a bijection? — ssu

I don't see any issue with bijection in principle. But when it is proposed that the quantity of a specific set is infinite, bijection would be impossible. The proposal of infinite sets presents numerous procedural problems. That is self-evident.

People can produce whatever axioms they like, but if they are not useful they will not be used, nor become conventional. — Metaphysician Undercover

Yes, that's precisely right, and is why your talk of axioms being "false" is nonsense. Axioms aren't truth-apt; they're just either useful for their purpose or not. And given that the axioms of ZFC are the most prominently used, it stands to reason that they are considered to be the most useful.

That's all there is to say about them.

It should be evident to any well trained philosopher, that set theory is just terrible philosophy. I think that is what bothered Wittgenstein about mathematics, but he was a bit too timid to actually come out and state it.

Regarding the "=" sign, it was invented in 1557 by Robert Recorde:

And to avoid the tedious repetition of these words: "is equal to" I will set as I do often in work use, a pair of parallels, or duplicate lines of one [the same] length, thus: =, because no 2 things can be more equal.

What????

Isn't there a bijection between the set of natural numbers and the set of natural numbers?

If so,

Isn't there also a bijection between the set of natural numbers and the set of rational numbers also? And a bijection between the set of natural numbers and the set of algebraic numbers? This is the reason why we have the "Hilbert Hotel" example and actually, the axiom of infinity in ZF as it's written.

I think there is as it's the way that set theoretic books describe it and the way I've learned Cantorian set theory.

That there isn't a bijection the set of natural numbers and the set of real numbers is basically why there is all the fuzz about aleph-0 and aleph 1. And here we get to the Continuum Hypothesis already.

Hence infinity is actually very puzzling to us. Still.

I think that is what bothered Wittgenstein about mathematics — Metaphysician Undercover

Was that early or also late Wittgenstein? Because I suspect late Wittgenstein wouldn't have read any metaphysics into set theory. It's just a useful language game we play, not something that entails the realist existence of abstract mathematical objects.

That's why we decided to construct formal systems with prescribed definitions and axioms to ensure that our maths was consistent. — Michael

Big problem with consistency when the use of "=" is not consistent.

Yes, that's precisely right, and is why your talk of axioms being "false" is nonsense. Axioms aren't truth-apt; they're just either useful for their purpose or not. And given that the axioms of ZFC are the most prominently used, it stands to reason that they are considered to be the most useful. And that's all there is to say about them. — Michael

In the sense that axioms are a representation of what mathematicians are doing, they can be judged as true or false, just like any other description. However, as you rightly describe, a judgement of the truth or falsity of an axiom is not required to judge whether it appears to be useful or not.

So this is where self-deception enters the environment. If a mathematician accepts an axiom because it is useful, but it is not representative of what that individual is doing mathematically (and this I argue is the case with the axiom which makes the claim about the relation between identity and equality), then the usefulness of that axiom must be in relation to something other than mathematics. It has some other purpose than a mathematical purpose.

Regarding the "=" sign, it was invented in 1557 by Robert Recorde:

And to avoid the tedious repetition of these words: "is equal to" I will set as I do often in work use, a pair of parallels, or duplicate lines of one [the same] length, thus: =, because no 2 things can be more equal. — Michael

Notice "two things". Equality deals with two things, identity only involves one thing.

Isn't there a bijection between the set of natural numbers and the set of natural numbers? — ssu

That's a bijection which cannot be carried out, cannot be completed. It's a nonsensical proposition.

Was that early or also late Wittgenstein? Because I suspect late Wittgenstein wouldn't have read any metaphysics into mathematics or set theory. They're just a useful language game we play, not something that entails the realist existence of abstract mathematical objects. — Michael

Notice early Wittgenstein talking about representing the world in terms of "elements". Notice later Wittgenstein rejecting this as not representative of what is really the case in the world.

Notice "two things". Equality deals with two things, identity only involves one thing. — Metaphysician Undercover

In the context of maths, when we say that A = B we are saying that the value of A is equal to the value of B. The value of A is equal to the value of B if and only if A and B have the same value.

A non-identical but equal value makes no sense.

We’re not saying that the symbol “A” is identical to the symbol “B”. This is where I think you are misunderstanding.