Philosophy of Mathematics

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The connection between math and natural language is one I’ve never heard someone flesh out like that. That’s an interesting point. Math certainly is a language in some sense of the word.

I’d say you’re losing a bit if the distinction between the words and symbols by which we express something, and the thing itself. Mathematical objects exist in that they are there for us to find and then describe with the human language of math. I have no problem admitting that “addition” doesn’t exist in the sense that we could’ve called it something, anything else. But addition, itself, in essence, is a truth that is outside of our recognition or naming of it. We didn’t invent addition to describe how turning two piles of things into one pile of things works. The piles had to work that way, and addition was already there for us to discover.
 
If you want to claim that than you have to claim that truth and love are the same thing. Love=God=Truth. I’m saying that doesn’t hold. So God contains more than one thing, but none the less I not composed of parts.
 
I’m not going to debate people who refuse to state their position. Good day to you.

(And no Catholics are saved by grace not intellectual gymnastics. There’s actually no intellectual requirement on me by the church. Fasting, mass, confession, Eucharist. Those are the requirements.)
 
Don’t get me wrong man, @chessnerd321. Back when I was majoring talking about epistemology would be practically dismissed - impatiently so. So I’m really sorry if some of that bad habit rubbed of on me.

What I was trying to say: Within catholic heritage there exists a tradition addressing the classic example in your OP. If you find the article off-putting, well I guess almost everyone would. I’ve seen that paradigm question made and answered so many times I can’t really think about.

But find value in engaging the person on the other side 🙂 even if I can’t answer.
 
The connection is between a formal language - that is, one that we create by stipulating the syntax and semantics (logic is such a language) and natural language - which came about organically.

I don’t know if there is a distinction between a symbol and a word, in that a word is a symbol. “Dog” is a symbol I speak to represent my little pug when I say “my dog is the cutest in the world.”

But it’s a foregone conclusion that we don’t find circles in nature - we find circular objects for sure. A circle is a mathematical object we invented to describe the circular objects we find in nature. And in a very real sense “circle” has always existed in that a logical object (as in, not physical) that is a line whose points are all equidistant to a central focal point has always been a truth, even without people around to see it.

Addition isn’t something we invented but, in terms of math, is it an operation indicated by the operator +. And it works exactly as other operators in logic do - and, or, if>then, and the biconditional. (and many more). But it’s also a foregone conclusion that’s been discussed long and hard that logical operators are imperfect - they don’t exactly capture what goes on in language, or, reality. They are models. 2+2=4 is not the same as bringing two groups of two apples together to make a group of four apples. It doesn’t even correctly describe the state of affairs, as it loses a lot in the abstraction from apple groups to a proposition of mathematics.

Now, I’m not a mathematician (though phil. math had be really considering going back to get another degree in math so I could be) so I can’t adequately lay out the math-y side of it all, so please forgive me. But the rub goes down to even simpler than we’ve been discussing, to the foundation of arithmetic. That is a puzzle that has been confounding philosophers for thousands of years and there is a great deal of contention about why arithmetic works. Mostly, we just ignore that contention because it does APPEAR to work - in works perfectly to the vast vast vast majority of people’s eyes.

But for me, the ontological problem, the semantic problem, the epistemic problem, the incompleteness problem all seem to be best explained by taking math as a form of logic as I’ve described.
 
chessnerd321 said:
  • Assume truth exists. (I’m not really prepared to argue about this one for the sake of staying on track.)
  • Let T = {A,B,C,…} be a set of all true statements.
  • If we can observe that the elements of T follow some basic laws, then the laws are also true.
  • Since true, these laws are in T.
  • The laws exist because they are true.
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I do not want to argue about your assumption, i just want to know what it means. Existence exists? Statements about existence that are true exist?
Statements exist if they are true? I am not at all sure what you are trying to say, and think this is the major sticking point.

Is T a set of statements which are all true? A set of all statements that are true? Or the set of all possible true statements? Different conclusions come from each of these. The 4th line is correct only with the last meaning.

Statements exist, true or false. If something is stated, its existence does not depend on whether it is true or false. But I do not know what you mean by “truth exists.” so I do not really know what you are trying to say.
 
Laws are in reference to reality that we experience. The accuracy of the predictability of these applications become a “Law”. But “Truth”, “Law”, etc are all words we use to describe our logical systems that reference reality and it’s predictive nature. If reality wasn’t predicable, then we would have randomness where one day gravity works the other way, snow flakes form in furnaces, etc. The fact that there are specific laws that force processes once these conditions are met is how we get evolution, snow flakes, and every other predictive process in reality. Truth’s are just facts in reference to reality. Laws are just the most predictive processes in reality given the unchanging truths of reality.
 
Asking if mathematics has any existence outside of physical reality is like asking if the game of chess has any existence outside of physical reality. Yes, we play chess with physical pieces, but those physical manifestations of chess pieces are just place holders for the real concept, which is the rules of the game. However I don’t think it is proper to use the word “exists” to describe them. They are pure concepts. Do they exist separately from the mind of man? I suppose you could say they exist in the mind of God, so even before there was man, God perhaps “knew” the potential of chess as a game. Some math is just like that. It is an abstract set of arbitrary rules by which mathematicians play. Of course a large body of mathematics is applied and does describe or model the physical world. In a sense, those branches of mathematics may have more claim to “existing” than purely abstract math.
 
I think I am more persuaded by Edward Feser’s book on 5 Proofs for God’s existence to think that mathematical objects like numbers exist objectively apart from any human mind but not apart from God’s mind. In other words along the lines of Aristotelean and Aquinas realism but not Platonic realism. In Platonic realism numbers would exist as eternal forms on their own (apart from any god) . The problem with this is that even Plato showed that these forms require their own forms to account for their existence. And those forms require further forms and so on to infinity.

However, in Aristotelean realism these abstract objects are still objectively real apart from any human mind but their existence is grounded in the divine mind.

And the problem with thinking forms and other abstract objects like numbers exist in only human minds is that it can be shown that these abstract objects like numbers would still exist even if there is no universe or human mind to contemplate them. 2 + 2 = 4 is true regardless if any universe existed.
 
Would 2+2=4 without God? Or to put it another way, could God make 2+2=5?
 
I think it was Godel who said: “God gave us the set of natural numbers, everything else is man’s invention”. Doesn’t help the discussion, but it just popped in my mind.
 
oldnskeptical said:
Where there is math, there is reality, and where there is reality there is math. Neither one preceding nor giving rise to the other.
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This is verifiably not true. As I and other commenters have said, this runs into a brick wall when it comes to esoteric math.

There is math, derived from the math which represents physical reality, which has no possible actualization in physical reality. This esoteric math is as true as the math it comes from. It’s proven beyond a doubt from the same axioms. So it’s either all real or all not real.
 
niceatheist said:
Would 2+2=4 without God? Or to put it another way, could God make 2+2=5?
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I should correct myself and say I am persuaded towards scholastic realism which is almost the same as Aristotean realism except that it can deal with the problem better since it accounts for the possibility of there not being a material universe to begin with then accounts for the existence of material objects and abstract objects through the divine mind.

Aristotean realism says abstract objects exist only in the objects themselves and in the human mind. But this does not account for the possibility of the universe or humans not existing or before their existing. What accounts for their existence then?

Platonic realism suggests these abstract objects live in their own realm all of their own. But there are problems with this view. The best view seems to be scholastic realism.

If we defined God as merely another being in the universe that has existence, like Zeus, albeit a more powerful one, then I would be inclined to say that 2 + 2 = 4 exists apart from him.

On the other hand if we define God as classical theism does, as being existence itself, then everything else has it’s existence through God. Since God is the only thing that has existence as his nature. There can be in principle only one thing that has existence as its essence. Everything else has it’s essence and it’s existence distinct from one another.

BTW conceptualism says that abstract objects exist only in the human mind. While nominalism holds that abstract objects do not exist at all. Both of these have their problems as well. What view do you hold?
 
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I was just contemplating this today and I would argue that 2 does indeed equal 1 mathematically. Mathematicians do not know what to do with this fact. But what he showed you about 1 = 2 when divided by 0 is the kind of chaos we can not calculate in the generation of a cloud. It’s a shame man shys away from his problems.

The problem with -1 divided by zero is understanding that the absolute value of -1 is 1. This is terribly useful to know here. There is so much more to say but this has a lot to do with my disagreement with Cantor. I’ll write more later.
 
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No, God could not make 2 +2= 5, and saying that does not deny his omnipotence. Just as God cannot take a true statement and make it false.

As Aquinas said " there does not fall under the scope of God’s omnipotence anything which implies a contradiction."
 
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This is a great conversation and I disagree I hypothesize that there is a mathematics which is logical and another mathematics within the same number line which is completely chaotic and I have evidence.

Consider the following:

X = there is not truth
Y = there is truth

If x=false x=y
If x=true x=y because x=true

So about how God can not take a true statement and make it false???
 
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catholicray said:
Mathematicians do not know what to do with this fact. But what he showed you about 1 = 2 when divided by 0 is the kind of chaos we can not calculate in the generation of a cloud. It’s a shame man shys away from his problems.
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Oh boy, we do no such thing. You just haven’t been taught it. Division by zero is defined in a set of mathematics called Wheel Theory. Look it up, you sound interested.

The problem is that this generalization causes a loss of generalization for other properties. Such as the distributive law, the existence of unique multiplicative inverses etc. While it is useful to look at algebraic wheels and see what is true about them, we can prove a lot more if we stick to fields. That’s why people prefer to use them.

It’s wrong, flat out wrong to say “you can’t divide by zero” but mathematicians say it all the time. Why? Because the algebraic system you develop is much more useful. Dividing by zero is of much smaller utility than the distributive law, for example. But it would be more correct to say “you can’t divide by zero, if you want the distributive law to hold”

Edited to add that 1 =/= 2 but only because 0=/= 1 by definition. This is again due to a frustrating loss of generality would result. There are however number systems where 0 = 2 (modulo arithmatic). Or even crazier things like apperently infinite numbers being equal to negative numbers (p-adic numbers).
 
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