Math feels like discipline with no reward.

Post Reply New Topic

I disagree on mathematics being "made". You may also be confusing my use of the word "God" as being related to a persona associated with religion in my world the word "God" is a way of describing the beauty of knowledge (I am fairly certain there is no word that would accurately describe what I mean).

As far as anyone knows, humans are the only beings on earth that formulate and create mathematical concepts and systems. Mathematics is an artifact, not a natural happening.

If there were no sentient beings in the cosmos there would be no mathematics in the cosmos.

ruveyn

Sorry, I just have to weigh in on this, since the philosophy of maths is an interest of mine!

To me, mathematics is all about proofs. A proof is a list of strings of symbols, each of which is "deduced" from previous strings of symbols according to certain rules. All of this can be made absolutely precise.

I believe in the existence of proofs, but I don't necessarily believe in the existence of the things that proofs claim to be about. So for me, we should never say "Mathematical fact X is true". We should instead say, "The statement X can be proved in the system P".

If you think that mathematical objects exist, and mathematical statements are really true or false, then there are a lot of philosophical problems you have to deal with. For example, are the axioms of ZFC (the most common foundation for mathematics) true? Does it matter whether they are true?

This argument is exactly equivalent to saying that the universe does not exist without humans here to observe it. I have been known to go as far as to say that the universe does not exist without ME here to observe it partially because the universe I live in is distinct from the universe you live in. We cannot individually separate our perception of the universe from the actualization of the universe.

The question here is is mathematics the language used to describe a thought, the thought itself, or something inherent to the fundamental structure of reality that we merely describe. Whichever perspective you take you are correct as truth is relative to the observer under a given set of assumptions which we cannot ever understand.

Let us take the set of all possible assumptions (axioms/basics) take any consistent subset and you have a predetermined set of logical conclusions that can be made from this subset.

I agree. However, that doesn't help with our problem of whether or not a given axiom is true.

For example, there is an axiom of ZFC which can be translated as:
"There is a set X such that, for any set Y, Y is not an element of X."

How can we decide whether this axiom is true? What is a set? Do sets really exist? How can we know that there is a set with this property?

I claim that this axiom is neither true or false. It is just an axiom. And I claim that the mathematics that is built up from such axioms is neither correct not incorrect. It doesn't describe any "real mathematical truth".* It just is. It's a game.

*not to be confused with applications to physics

Here we may contend with differing definitions of the word "true" or "truth" I claim that EVERY axiom is true in that there exists a situation such that assuming said axiom along with a few others creates a language for describing some thought.

I think so. For me, a claim can only be true or false when it refers to objects in "reality".

So for me, mathematical claims cannot be true or false, because there are no mathematical objects in reality for them to refer to.

However, metamathematical claims can be true or false, because they refer to proofs, and proofs exist in reality.

e.g. "There exists an empty set."
-mathematical claim
-neither true nor false

e.g. "In ZFC, the statement "there exists an empty set" is provable."
-metamathematical claim
-true

I claim reality can no more be "true" or "false" than mathematics. A statement can only be true or false because it follows from the assumptions made by the observer. I give the simplest argument I can: Is it true that YOU exist? I assume that I am interacting with "reality" (not a "true" statement according to your philosophy but it is an axiom most live by.) That along with many other assumptions leads me to believe that indeed you exist is this truth?

From this I can make the claim that mathematics is more true than the reality we live in in that it makes explicit the assumptions that are made to come to the conclusion.

I thought that the problem was that I was more cynical than you, but now I see that the problem is that you are more cynical than me!

I may not believe in mathematical objects such as sets, but I do believe in reality. I just.... do. Don't ask me to prove that reality exists, I don't know how!

You can't the furthest anyone has gotten is "I think there for I am". Everything beyond that is an axiom or a conclusion from said axioms. I simply state that the concept of "truth" is most fully justified when the assumptions made to come to the conclusion are as fully stated as can be.

To be clear I take reality as an axiom as much as the next man I am just aware that it is indeed an axiom. I am also a groupie of truth I just take truth to be that truth "a" is consistent with the rule set "b" coming from the axioms of the system I choose to perceive.

By the way thank you very much for the conversation it has really brightened my day. It actually gave me motivation to think.

Me too!

It's not very often that I get to talk about the philosophy of mathematics with someone.

Fractals are indeed awesome.

Writing code for that is actually not very hard. Do a bunch of iterations of z = z^2 + c (where z and c are complex numbers). c is the complex number of a grid position in the complex plane, and z is 0 initially. If you go from -2 to +0.5 on the real axis and -1.25 to 1.25 on the imaginary axis, you should get the whole set on the screen. If z goes to infinity (if it goes farther than about 2 units from the origin) then color the pixel corresponding to c with a color that depends on how fast it left. If z doesn't go far from the center in a set number of iterations (256 is a good one to start with), then it's part of the set, so color it black.

There are optimizations that you can make to avoid excess work, but that shouldn't be too slow to start off with.

There are actually a lot of fractal software programs that contain their own language and compiler that will do high precision floating point math on complex numbers very efficiently. Using 128 or 256 bit floats allows you to zoom to depths that you can't get with the normal 64 bit double precision numbers. You can also tweak the iteration algorithm to get all kinds of different fractals and color them any way you want using creative mathematical formulas and coloring/shading gradients. I really got into making fractal artwork about 8 years ago. The software's probably gotten even better these days. The program I used cost around $40.