Applicability of Maths to the Universe

[polymath257]

Yes, there is an aspect of math that cuts across cultures. But this is also true of other basic linguistic concepts. So, cat, chat, gato, mao, etc as opposed to two, deux, dos, er. 

The word "cat" and its cognates refers to material objects of a certain type. The question we're working on now is: what does "two" refer to?

What does 'pain' refer to? What does 'abstract' refer to? What does 'love' refer to?

They are *concepts* we have with no external or material referent. The number 2 is like that. It is a placeholder, not a thing in itself.

One difference is that math  is a *formal* language: it has internal rules that are not present in most natural languages. And, for mathematicians, playing with and exploiting those formal rules are the essence of the game.

And, yes, mathematics really is like a very complex game for those doing mathematics. It has rules about what 'plays' are legal, it has goals (theorems), etc. It can even be helpful to *think* of the mathematical concepts visually and in other ways.

You use words that sound unserious when you talk about pure mathematics -- "playing with," "game," "plays," etc. 

Would you say that math is only serious when it is used to describe the material world? That any other time it's just a game? 

I think that many mathematicians would disagree with you. I'm also concerned that if we define seriousness as utility, we're repeating a common anti-intellectual assumption.

In case you haven't noticed, people can take games very seriously.There are serious chess players, serious athletes, etc. That doesn't mean they aren't ultimately playing a game.

Mathematics is a very challenging intellectual game. It has puzzles in it, challenges, exercises, etc. But in and of itself it says NOTHING about reality, except that the game can be played.

Now, it does turn out to be a *useful* game, but like you said, that is a different matter. Whether that utiliy is serious or not depends on the user.

In exactly what sense do numbers have an 'independent existence'? 

Since you've already ruled out all of Plato (without bothering to explain why), we can talk about numbers having independent existence in exactly the way that Popper describes. I suppose I could type that all out in my own words, but the original paper is not long. And if you're going to go around declaring that Popper is wrong it might make sense for you to read what he says. 

https://tannerlectures.utah.edu/_documen...pper80.pdf

What I find most interesting is that Popper *doesn't* claim that world 3 objects have an independent existence. They are created by humans for human reasons and to do things to and for humans.

And, like the world 2 objects, I think they ultimately supervene on the physical. They are *concepts* in human minds, common actions or beliefs across humans. They only exist because we exist thinking about them.

From what I can see, the 'number 2' is a shorthand for all the cases where counting two objects is a useful thing to do. And the mathematical object 2 allows for such modeling.

So a number is the memo we use after looking at two objects and counting them? If that's all numbers are, then you're begging the question and assuming that they only exist when used in reference to physical objects. 

No, I am saying they only exist as concepts in our minds. They have no independent reality. We can, for example, use them to count thoughts or imaginary pink unicorns.

But then, you use the term "mathematical object 2." Is this the same as the number 2? Do people doing pure mathematics use "mathematical objects" but not numbers?

I would guess that very few ordinary people think of the number 2 as the set {{},{{}}}. They are more inclined to think of 2 as being the successor of 1 in some inductive set, but I doubt they think of it in those terms.

Both of the mathematical formulations are based in the specific axioms of set theory used for the development of modern mathematics.

And, in most ways, the second sense is the closest to an actual definition of 2. You start with an inductive set: just a set with a one-to-one function and a special point (depending on the formulation, we call that special point 0 or 1). if we call the special point 0, we define 1 to be the image of 0 under that function. And, in either case, we define 2 to be the image of 1.

But what this means is that there are *many* versions of the number 2: one for each inductive set. They are considered equivalent because equivalences between such inductive sets.

The average mathematician would find this rather obvious after a bit of thought. But I very highly doubt that the average person has ever thought in these terms.

In other words, there *is no number 2* that is different from every other mathematical object. The property of being 2 depends on the specific system it is in (the inductive set). And it is a matter of definition (does our inductive set start with 0 or 1?)

Now, the first definition of 2, as {{},{{}}}, is useful in math because it gives a very simple example of a set with the required properties for a very easily defined inductive set (the ordinals). But, again, I very much doubt that the average person thinks like that.

On the other hand, Frege's definition of 2 as being the collection of all sets of pairs turns out to be inconsistent. That collection cannot be a set because of Russell's paradox (ultimately).

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I really want to see some one counting in 'Cat'!

  

No one counts in "cat." 

The question is: does a word like "two" refer to an object in the same way that a word like "cat" does.

No, it doesn't. But neither do words like 'thought' or 'good' or 'beauty'.