Philosophy of Mathematics

#141

You’re notation is inconsistent. And I do not mean the inconsistent use of upper and lower case, I am assuming it is case-insensitive, ie x is the same as X).
What I mean is that you define X and Y as statements. And then you have a condition on those statements being true or false. So far so good. But then you redefine X and Y, effectively treating then as mathematical variables. The whole thing then does not make sense.

I go back to this post of yours from yesterday to illustrate where you are getting off base. Your ideas are worth considering, but you have some basic misunderstandings about math which are confusing the matter.

#142

Ok, let me try to put this in the correct terms and then provide a conclusion. First of all, you say “math is objectively true and objectively false”. This seems to be equivalent to how a mathmatician means “a mathematical theory, such as number theory, is inconsistent”. Now, by theory, we mean a set of axioms and a given set of objects. Examples would be simple number theory or set theory. What does it mean for a theory to be consistent? It means there is no statement in that theory can be proven to be both true and false. If a theory is not consistent, it is not very useful. It turns out that consistency is very hard to prove. As a matter of fact, the only very simple mathematical sytems (theories) can prove themselves to be consistent. Any theory, at least as complicated as number theory (Peano number theory) cannot be proven, with only its own axioms, to be consistent. But that does not mean that the theory is inconsistent.

#143

Those times when math attempts to describe the real world, it can be said that the math is either objectively true or objectively false. But when math is describing a made-up world (which is often), there is no sense in which the math is objectively true. The only thing you can say about it is that it should not break the rules of the made-up world to which it applies. You can have two different made-up worlds with two different mathematical theories. A true statement in one of those worlds may not be true in the other one. This is hardly an “objective truth.”

#144

I would argue with your “often” statement. Yes there are some rather esoteric math systems which do not seem to reflect reality. But in general, the vast majority of math theories do. And the term objective is redundant in my mind when dealing with math. What is true in math is true in reality. Euclidian geometry, number theory, set theory, calculus, algebra, arithmetic, discreet mathematics, theory of computation, differential equations, statistics, linear algebra, formal logic: all are types of math that I have studied at one point or another in my life. They all are true. It would have never crossed my mind to think that these subjects are not objectively true.

#145

Truth is the conformity of the mind with reality. If you are dealing with mathematical objects that are real, then there can be objective truth depending on whether your thinking agrees with reality or not. But if you are dealing with esoteric mathematical systems involving derivative mathematical concepts, or objects that have no existence in reality, then you can’t speak of “truth” because there is no reality for the mind to conform with. However, you can still judge whether the system is logically consistent or not.

You are right, most esoteric systems are of interest mainly to pure mathematicians. However, some of them, such as Riemannian geometry, have been used successfully by physicists in the study of the physical world. Is Riemannian geometry objectively true? I will refrain from putting it that way. But I would not hesitate to say that it is a consistent mathematical system useful for studying the physical world.

#146

I don’t think that math is something that is created nor discovered.

Math is a way of explaining the underlying properties of the universe. Over time, individuals have generated theories about those properties. Over time, those theorems have been tested by others, and we have come to accept them until someone else can prove them untrue.

If no one ever theorized that the hypotenuse of a right triangle could be determined by the other sides, it would still be true. There are undoubtedly many such facts out there that no one has yet pondered.

So what you say is math, is just an accumulation of all such theories. Those theories are simply our explanations of truths of our universe.

Let’s take math out of this for a minute. Let us say that you are one of the first humans ever to exist. You observe a rainbow. After a while, you begin to notice that there are often rainbows during or after rain showers. You begin to notice that the rainbows always occur with a certain relationship to you and where the sun is. You also begin to notice that the colors are always in the same order. You are excited about your observation. You would like to share it with others, but language has not yet been developed. First you will to somehow identify each color. Maybe you take leaves, berries, rocks and other things that represent each color and place them in a pattern on the ground. Next time, it rains, you observe that the colors are the same as your pattern. You share this with another human by pointing to the ground, and the sky until the point is made.

It seems to me that the explanation is like math. You made an observation. You aren’t able to prove it is true, but every time you observe it again, you are assuring yourself that it might be true. It will only take one time for it to be proven that you are wrong, but until then every time strengthens your argument.

Have you created or invented anything? (No) Do the colors exist (Yes). Does the pattern exist (Well, in your observation, Yes). Did God create the colors? Did God create the pattern? Did God purposely create the pattern, or was it an inadvertent result of some underlying physical properties?

#147

That’s just not how proofs work. Like at all. Proofs are firm and logical. There’s no room to be wrong. Is it possible you’re getting mathematics and mathematical laws of science conflated?

Additionally it sounds like you think math is discovered:

That’s almost exactly what I mean by discover. It’s true whether we find it or not. When we find it we’ve discovered it.

#148

But if we take the words “objectivly true” literally then it obviously is. Riemannian geometry is provable within a wider mathematical framework part of which describe the physical world. It doesn’t make sense to say only part of the results of an axiomatic system are true. It’s all or nothing, just as a matter of logical consistency.

#149

What you say might be true for applied math. But here are few fields in mathematics that do not have their roots in the physical world, even if some of them do have real-world applications.

1. Graph theory
2. Combinatorics
3. Galois theory
4. Neofields
5. Finite projective planes
6. Elliptic Curves (as in Cryptography)

There is no sense in which these things existed, except as a potential concept in the mind of God. They did not describe anything in the physical world.

#150

Proofs are firm and logical, but they must be based on something that is not proved. We hold these as accepted truths. Once one sets up the underlying accepted truths, then the logic flows from that. Given A and B, then we can prove C. But what if someone later can prove that A or B are not always true in every situation?

(I have to admit that it has been a long long time since I did my undergrad)

#151

With regard to Riemannian geometry I said that I would refrain from calling it “objectively true,” not because I doubt its validity, but only because some of the objects it deals with are purely mental constructs and do not exist in reality. For example, objects of 4, 5, or 6 dimensions aren’t real. Are the propositions you make about them true? I guess it depends on how you define truth. But I defined truth as the correspondence of the mind with reality. So, if an object isn’t real, how can there be truth? If an object isn’t real, then the propositions I make about that object could be good or bad, right or wrong, valid or invalid, - depending on whether they are logically consistent with the definitions and axioms of the system - but it is meaningless to call them true or false.

Some of the theorems of Riemannian geometry also apply to real objects, such as 3-dimensional objects or surfaces. Mathematical theorems you make of them can – according to the definition of truth that I started with – be meaningfully called true.

#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.

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