Mathematics - Knowledge About the World or True By Convention?

[JasonM]

There are two sides to the debate about the philosophical nature of mathematics. One side claims that math is something that is out there. For example, if we were laying tiles in a rectangular room, and we figured out that we needed 12 tiles along one wall, and 12 tiles along a perpendicular wall, then we would need 144 tiles. There is something interesting about this. We feed in the appropriate data about the room, and math spits out the right answer. In other words, math is telling us something about the external world. This is the view of the mathematical realist.

To a mathematical conventionalist, this is not interesting. By multiplying 12 x 12 and coming up with 144, we are essentially saying the same thing. In other words 12 x 12 = 144 is just a convention that we invented. It tells us nothing about the external world. It would be like saying that a bachelor is an unmarried man. This statement doesn't help us discover anything about the outside world, it just states something that is true by convention.

Mathematical realists run into a problem in that mathematical knowledge isn't obtained by the five senses. Our five senses are our only window to reality. That's how all sciences gain knowledge, so if mathematics isn't determined by our five senses, how can it be something that exists in the real world?

On the other hand, mathematical conventionalists also run into a problem. For example, an alien species can come up with an alternative to math, schmath. In schmath, 12 x 12 = 150. In fact, all the other rules are different. The problem with this alien species is that schmath will lead them to build bridges that fall apart, machines that don't work, and aircraft that run out of fuel before they reach their destination. This shows that there's more to math than mere convention, because if it was based on convention, it wouldn't matter what the rules are.

I think I know what the problem is. Mathematical knowledge is not knowledge which is based on experience, it is meta-knowledge: knowledge about knowledge. Formal logic is in the same class. For example, when you say A =/= not A, you are talking about knowledge you have about your knowledge. You are saying that anytime, in any case, if you have a statement about one thing, it is not logically equivalent to it's opposite. This is similar to what mathematical knowledge tells us about the world. It is something that is out there, because it is like empirical knowledge; there are still many rules about our knowledge that we have to discover. It isn't merely a convention, because the rules to our knowledge aren't just conventions; they have to be learned and discovered. Therefore, I'm on the side of the mathematical realist.

What's your opinion? (I know that this is a very technical issue, so I hope I was clear about the concepts involved; if there's something you don't understand, maybe I can help clarify it.)

[Monte314]

Jason--

An interesting question, and quite technical as you suggest. It is most often stated colloquially by asking, "Is mathematics CREATED, or is it DISCOVERED?"

19th century mathematician Leopold Kronecker said, "God created the integers; the rest is the work of man." For him, mathematics is not "out there" in any absolute sense.

Modern constructivists/intuitionists/finitists insist that mathematical proofs that include processes that cannot actually be carried out *in principle* are invalid. This stance calls into question many fundamentals that others take for granted (such as the existence of the real numbers, the existence of unmeasurable sets, etc.) In this sense, constructivist mathematics is "more likely" to just be "out there". A lot of mathematicians tend toward the "out there" side. They are like the sage who felt that he was "thinking God's thoughts after Him". To them, mathemcatics is about discovery through creativity, but not about creation.

Leibniz, co-inventor of aspects of Calculus with Newton, took a middle of the road stance: he felt that mathematics was "out there", but in such a way that the observable world conformed with its principles by a sort of enforced coincidence. (He attributed this to the operation of fundamental entities called "monads"). See his book, "Monadology".

But some resolution is now possible. In the middle of the 20th century, it was shown that distrinct, yet completely valid versions of mathematics exists. Further, there is no objective reason for preferring one over the other. In other words, which "version" of mathematics I regard as "true" is a subjective choice that can be made for any reason... or no reason at all. An excellent treatment is the book "Mathematics: The Loss of Certainty" by Morris Klein.

For example, I can choose to accept the Axiom of Choice as true, and I get a consistent mathematics (which is the arbitrary choice made by many mathematicians). Or, I can choose to reject the Axiom of Choice, and I get a consistent mathematics (which is the arbitrary choice made by many mathematicians). These distinct version of mathematics have a lot of overlap, but they disagree on some important theorems. What is "true" in one can be "false" in the other. Mathematical "truth" is now known to be contingent upon subjective choices.

Who is right? Both... neither. Flip a coin... and if you want, change your mind after lunch.

There are other arbitrary choice points in mathematics... FAITH in the Continuum Hypothesis is one... FAITH in the Extended Continuum Hypothsis yet another. The list goes on.

All of this suggests that if the universe is based upon some underlying absolute formalism, there will nevertheless be valid versions of mathematics that are inconsistent with it. Since it probably isn't correct to refer to the "discovery" of something that isn't there in the first place, the "discovered" position wouldn't apply without qualification.

Personally, since I believe in an omniscient God, I must conclude that whatever "mathematics" He "knows" is True, and potentially discoverable, as the first chapter of Romans (New Testament book by the Apostle Paul) suggests. Other versions of mathematics, would then, be "created" (concocted!)

[cha071c0rd3r]

Mathematics, in my opinion, is about creating from a foundation. The foundation is what we observe around us. "I observe that I can count the number of objects in front of me. I observe that this number is not fixed (i.e. I am able to count 2 or 5 or 10). I conclude then that this thing I call the "number" is a comparative quality of the physical world that I am capable of perceiving." This then can be extended to abstract ideas and we no longer depend on physical objects in front of us to induce the idea of number.

This is the part where we take those simple observations of what is true and follow a thread of logic to arrive at other facts. We see that is there is 4 of something, we can count 2 and then 2 more, or we can count 1 and then 3 more, or we can count 3 and then 1 more, this is why 2+2=4, 2+2 is the SAME as 4. 12x12=144 by definition of the operation, and it cannot equal 150, because 12x12, by definition, means take 12 12 times, if you make 12 rows of 12 objects, counting one by one you will get 144.

Notice the word definition, that forms the basis of mathematics. Once we accept this concept called "quantity" we use our imaginations to give arbitrary names to all kinds of quantities, and then see what we can derive from the definition of the quantity (take derivatives, for example, which have a very precise definition: it is not that Newton and Leibnitz discovered those things called integrals and derivatives; what they did was "invent" a certain meaningful quantity by mixing certain other quantities which were already meaningful, and then seeing where they could take it).

The truths obtained through mathematics are in a fundamental sense self-contained. That is, if we change our assumptions, question the fundamental axioms, the results will be false within the new system. However, as long as you accept the foundations as true (like, you agree that addition and subtraction are possible, and indeed, if you have 3 objects in front of you and you take away 1, you will be left with 2; and you agree with the language, e.g. you know what is meant by the word "add" and you believe that it is possible to perform that operation, and you know and understand what is meant by "3" or "8"), the results that you get will be true. Since, once more, mathematics is nothing more than making logical conclusions from fundamental assumptions.

Thus, mathematics is more of creation, not discovery. It is a tool for discovery, however, as it allows us to learn things more quickly about the world, to make predictions based on a well-organized logical system - which is what math is in the first place, a way to organize the quantitative concepts we are aware of in a form which we can effectively use.

[errrzarrr]

Oh maths! I love it.

[AntimonyLegault]

Math is still missing the number between 9 and 1+0, that's the whole thing with precision and accuracy in physics. Sort of like the calendar and leap years.

[sriv]

Math is a language by which we describe the world. I think the basics of the language of math were discovered. The rest was/will be created by speculation and proved.

Interesting point is that different people percieve the language of math differently. Just like N has trouble with language and solidification, savants go about math differently. I saw this show where the savant described his rapid calculations as pictures swirling together and popping out a number with a unique feeling associated with it.

In short, the human brain has a limited view of the beauty of mathematics so we may not know all the possible conventions or knowledge about the world it may contain.

1000th post...the number gives me a gleeful feeling. lol, jk.

[cha071c0rd3r]

Interesting point is that different people percieve the language of math differently. Just like N has trouble with language and solidification, savants go about math differently. I saw this show where the savant described his rapid calculations as pictures swirling together and popping out a number with a unique feeling associated with it.


I believe that is called synethesia (when you mix up some of your senses).

[sriv]

I believe that is called synethesia (when you mix up some of your senses).

Does that imply that math can be sensed?

[cha071c0rd3r]

Does that imply that math can be sensed?

In a way, I mean you have to think of numbers in some way, no? When you think of "5," some abstract thought comes to mind that represents that "5." When senses that generally aren't used to process a particular object (abstract or not) leak into your understanding of the object, that is called synethesia. (Like feelings and numbers, or color and numbers, or size and numbers in this particular case).

[ayre]

This post will probably be moved elsewhere by the police who patrol this forum, as I seem to have a knack for getting off topic. But the connections to me are apparent. Chris Langan (not sure of the spelling of his last name), the man with the world's highest recorded I.Q., claims to have proved the existence of God mathematically. Now I say that is nonsense, because assuming there is a God, then S/he cannot be brought under the scrutinizing gaze of mere mortals. A God who has been seen through a telescope is no longer God.

My friend (an INFJ, but what does it matter except for the purposes of this forum),says that I am wrong. There are many "proofs" for the existence of God that are considered conclusive by those who believe in them, and a mathematical proof is no different from the rest. It will appeal to some and not to others.

Not being a mathematician myself, I'm not sure anymore. It seems to me that all the traditional "proofs" (cosmological, teleological, ontological, the argument from motion, and I think there's one more), go only so far. They don't claim to deductively arrive at God; instead, they appeal to an intuitive sense of what "must be" given certain observations. However, the entire process of reasoning is open to questioning, because it is philosophical in nature, not mathematical, and there are no final arguments in philosophy.

In contrast, math is a deductive science. Or at least that was my understanding. So if God has actually been "proved" mathematically, then we must all either believe in God or else throw out mathematics, which is sheer idiocy. There is no alternative. We must all believe in God. This is untenable. I conclude it is impossible that God has been proved mathematically, and more than this the claim itself is preposterous.

But now there's Zeno's paradox. Here is an example of a conundrum that does not admit a rational solution, which is the reason it's a conundrum. But the conundrum is mathematical in nature, so my friend says. His point is this: not all mathematical problems admit a clear cut answer. Perhaps. But Chris Langan doesn't say that his proof for God falls under this category. He claims to have actually proved God, i.e., arrived at an answer that can be mathematically demonstrated in a way that does not leave room for doubt. Moreover, Zeno's paradox is a logical problem, not a mathematical one. Or am I wrong?

[Adrian]

For me Math is a language to simplify complex things. So it is knowledge about the world, just simpler and handier to deal with, instead of gaining experience through empiric actions.
The conventions of math is not important.

ex.
You could say 2+2=5 if you assume that 5 is something else than 2 and 5 is something you could use to replicate this assumption. So mathematics is about replication, as well as every science. Since replication in science is equal to verification the conclusion is: math is true by every convention as long there is replication.


edit: you might want to read something from this: To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. (yeah, i know wikipedia sucks, but how about reading about philosophy of science instead).

[ayre]

So if we accept Christ Langan's definition of terms, then he has proved God, but we don't, then he hasn't. It seems this is what you're saying, unless I'm wrong, which I could be.

I know I'm way over my head with this, not being a mathematician, but it just struck me as odd that anyone would claim to have proved God mathematically.

[Adrian]

God and science shouldn't be brought in relation in a sentence imho. You can change your god, but you cannot change science, as it affects you. Well, although it depends how you think about it.

I've never read his publications, but think of it like this:

What is his goal? Why does he claim to know this? Are you able to totally falsify his statements under every circumstances? Is (or will ) he (be) able to verify his statements, if we all say no. Why does he have influence on your oppinion? Does he know that - of course he does.

Even if i didn't read his work, I already know what it says. And I believe most people here are thinking the same way.