Philosophy of Mathematics


#152

I don’t feel that you addressed my point. How is that something is true untill it no longer applies to physical realities? So a set of axioms prove things about physical objects, ok good. We agree.

Those axioms prove things which do not pertain to our physical world. Then it’s a stretch to say these proved statements are true? I don’t see how that’s logically consistent. It’s like saying A -> B is only meaningful if B is also a physical object. Hm. That seems deeply wrong.

Let me point out another reason I don’t think your view is consistent. There is no such thing as an actual physical sphere. Only things that approximate one. Riemannian geometry doesn’t exist even in 2 or 3 D in the physical world. So if statements about 4D objects aren’t true because the world doesn’t have 4 dimensions, then nothing in geometry is true, since no physical objects map perfectly onto mathematical ones. Thus we’re back to the same ol’ all or nothing.

Thus I say ALL of it exists outside a physical reality.


#153

It’s just because of the way I defined truth. Do you have a different definition? My definition of truth is the correspondence of the mind with reality. I can make up or imagine something that is not real, and my mind can have a perfect grasp of it, but that perfect knowledge will not fit my definition of truth. The definition I have given is the philosophical definition of truth, according to which truth and being are interchangeable. We are here talking about the philosophy of mathematics. So, I stick to the philosophical definition, which requires that truth be related to being.

Yes, it is a stretch. I can say that the proved statements are valid or correct. But I can’t say they are true. You can, of course, redefine truth to mean anything that is consistent, valid, or correct. In that case, you can say that all consistent mathematics, including those that have no bearing with reality, are “true.” But that is not the philosophical definition of truth.

The “sphere” is the conceptualized property of actual spherical objects which may not be perfectly spherical. In a way they are like the essences of things. When you see a child that is born with a defect, don’t we still say that the child is human? The essence of humanity still exists in the child even when the child has an imperfection. The essence “sphere” exists even in objects that are not perfectly spherical, otherwise the mind will not be able to abstract and form a concept of it. The real mathematical objects that I have been talking about are those that were derived from real things, not those that were derived from other concepts. Spheres, cylinders, lines, triangles – all these are real mathematical objects, whose concepts were derived from real things that are not perfect.

But you can also have concepts that are derived, not from real things, but from other concepts. For example, you can have a concept of a “four-dimensional surface” derived from your concept of a three-dimensional surface. Such “four-dimensional surfaces” are no longer real. Statements that we make of them may still be judged as valid or invalid, correct or incorrect. But from a philosophical standpoint, they are neither true nor false, unless you redefine the meaning of truth.


#154

In that case the entire argument crumbles, and new assumptions have to be made.


#155

Oh ok, you might want to let the philosophers know you found a definition for truth. I’m sure they’d be happy to know.

I kid, I kid. Mostly.

I can go with this. However: You’ll note that I was very careful to always say physical reality. I am absolutely not a physicalist. So no, non physical does not imply non real or non true to me. Based on the rest of your post you don’t sound like a physicalist either.

I’m not sure what basis to argue on anymore. Your arguement seems troublesome for so many reasons, unless I completely misunderstand and you’re a nominalist. But you sound platonist.

For example, your conception of ‘true’ implies that if I destroy every sphere in the world it’s no longer true that the volume of a sphere is V = 4/3×pi×r^3. I don’t know why it doesn’t trouble you that mathematical proofs which are the sheer results of logic alone are dependant on physical objects. I just still don’t see how that’s possible. Maybe we’re completely talking past one another.


#156

Some mathematical systems, especially if you talk about axiomatic systems, are surely countable. However, the world is not an axiomatic system, so when you talk about the whole world, you run out of the domain of the countable.


#157

I am neither a physicalist nor a Platonist. If I were to describe myself in terms of my philosophical outlook, then I’d say I am a realist.

If there was never any sphere in the world, you would never be able to make the statement that the volume of a sphere is V = 4/3×pi×r^3. But since there were actual objects from which you got the concept of a sphere, then the statement V = 4/3×pi×r^3 is true and will always be true even if suddenly all the spherical objects in the world were annihilated. Let me explain why.

Truth is the conformity of the mind with reality. But reality consists of actual beings and possible beings . Actual beings are those that actually exist outside the mind; possible beings are those that do not actually exist, but can exist outside the mind. The only beings that are unreal are those that can’t exist outside the mind, such as a four-dimensional surface in a three-dimensional world.

Now, let me go back where I left off. Even if suddenly all the spheres in the world were annihilated, the sphere would still be a possible being and, therefore, a real being. A planet of 1mile radius may not actually exist, but it would still be true to say that its volume would be about 4.2 cubic miles.

When you read my previous posts, and you see me use the word “real being” or “reality,” please understand that I do not merely mean to include actual beings, but also possible beings. Of course, we derive the mathematical properties of real objects only from actual beings, for we use our senses to perceive things and their mathematical properties. But once those properties have been derived and formed into a concept, the actual existence of those things are no longer required to make true statements about them.


#158

I was talking about statements in formal systems. However, regarding a natural language, if we assume that all sentences are finite in length, the the set of all sentences is countably infinite.
If I have some time, I will look up and reference the proof of this claim. I am working of long term memory, but I am almost certain I am right.

ETA: it may actually be finite, but again, I will have to do some digging.


#159

How do you formalize the world/ the creation? The topic, if I understand correctly, began with a very ambitious approach about thinking of God as a formal system that includes mathematics.

Look at this question:

How can you interpret a question like this?


#160

If you limit the alphabet and symbols used to 100 in the sentences, the monkey at the typewriter total for sentences of length L is 100^L
The sum of sentences of any length up to a finite number is going to be a similar finite number, F.
If f’ is the subset of F that are in natural language, f’<F.
If f’’ is the subset of f’ that make sense, f’’<f’.

So the number of statements is finite.
If the length of sentence is infinite, each length corresponds to a finite number, so it is a countable infiniy.

Since I usually leave something out, let me know what.


#161

I think the topic has been different. It started with a notion of a reality from which we abstract the formal system of mathematics. Is that formal system something we made up, or does it already exist in the reality/mind of God? What if the formal system describes a non-existent reality, like a 20 dimension sphere? (I am not conceding that 11 dimension shapes are not real, let alone 4 dimensional spheres which are certainly real though we cannot imagine them)


#162

I didn’t say you could. I simply stated (in the context of mathematics) that the set of true statements exited (if I remember how my participation in this thread started).


#163

I don’t know, its been so long since I studied all of this language structure stuff, I don’t remember all the details, but I am pretty dang sure that if there is an upper bound on sentence length, it is certainly not uncountable (sorry for the doublt negative)


#164

Can you give some further explanation, because im not sure what you are asking.

To me God is truth, And if anything else exists it is true because of God, and not a self-existing truth. In other-words it is contingently true. Without God, there is no truth because nothing would exist, which is, i think, impossible.

If there was absolutely nothing, it cannot be true that 2 + 2 = 4, because it is only truth because of the nature of that which eternally exists. It cannot be true because of nothing, because there is absolutely nothing in nothing, and something that is necessarily true cannot begin to be true because it is necessarily true. So there must be a reality that necessarily exists.


#165

That’s a misunderstanding of my OP. “Math is part of God” is not the same as “math can explain God”. Which is what you’ve suggested I said.


#166

A problem you are going to encounter right off the bat is whether God has parts. According to Aquinas, no. God is a simple being. Nothing can be a part of God. ( (Summa theologiae I, q. 3).


#167

I’m using informal speech to describe my views. If you want a full idea of what I mean you’ll have to follow the post. Most posters here have been a part of this discussion for a couple weeks now.

Sure God doesn’t have parts, but God cant be boiled down to one word descriptions either. God is love. God is Mercy. God is Truth. God is not only love or only Mercy or only Truth.

So God is math. Doesn’t mean “God is described by math” or “God is only math”. “God is not only math” is what I mean when I say “Math is part of God.”


#168

Just making sure that you are not confused. When discussion theology on a Catholic website, it’s best not to use terms loosely.

How much philosophy have you studied? If you don’t have at least the presocratics, Plato, Aristotle and Aquinas under your belt, your basically spinning your wheels dabbling in theological speculation. I’d also toss in Descartes and Leibniz, as well.

There is a vast body of literature out there on the philosophy of mathematics and its intersection with theology.

A good place to start is the following book:

Hacking, Ian. Why is there philosophy of mathematics at all? Cambridge University Press, Cambridge, 2014.

After that, read:

David Corfield’s Toward a Philosophy of Real Mathematics

Those will give you a good overview of the field. The second one is especially good, because it is written by a mathematician rather than by a historian of philosophy who doesn’t really know much about what math is.

Good luck!


#169

It does seem you are right, if a language has M words, and the maximum length of any statements is bounded by N, it does seem like the number of statements has to be less than or equal to M^N, which although is very, very large number, is certainly finite. But, I cannot help recalling that the number of statements in a formal math theory such is countably infinite. Furthermore, I recall from studying natural language processing that if the length of a sentence is not bounded (grammatically it is not, in the real world it is), then the number of sentences is uncountably infinite. So something doesn’t make sense. I would have to go back and do some research. And I don’t have time to do so.


#170

That is the conclusion I reached. For setnences of length N, the number of possible statements is s < M^N. That means the sentences of all lengths are a countable infinity.

Natural language processing is a whole nother world, of which I am completely ignorant. I am guessing our creation of upper bounds is a problem, since smaller units like letters and words are not single meaning units. But that is just a guess. I was too busy walking miles to school through 3’ of snow to fill my head with such ideas.


#171

Can you show how a 4-dimensional sphere is real?


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