Mathematics: created or discovered?

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The methodology of Euclid (however imperfect its execution) was the standard and model of mathematical rigor for nearly 2000 years. All of modern rigorous mathematics is an attempt to realize the ideal behind Euclid's work. Modern mathematics was born in Greece over 2000 years ago. The ancient mathematics to which you refer is of the Egyptian and Babylonian variety. Mathematics as a kind of inductive heuristic and not established by deductive principles. Not that there is anything wrong with Egyptian mathematics. I helped build the pyramids.

Here is a comment on what Richard Feynman had to say about the difference between Greek mathematics and Babylonian mathematics:

"In the second lecture, Feynman talks about the relationship between mathematics and physics. He says that mathematics is the language of physics but not the same as physics. He uses the metaphor of “Greek” and “Babylonian” mathematical traditions. In this metaphor, the Greek approach was axiomatic. Math is reduced to exploring the logical outcomes of a fixed set of rules or axioms. Everything connects to everything else within that system. The Babylonian method is to learn mathematics through examples and heuristics. The connection between these individual results may or may not be known. To Feynman, physics uses the Babylonian approach and it is only by doing it that way can new discoveries be made since theorems can pop out where they shouldn’t. He uses the example of the conservation of angular momentum, which was noticed by Kepler in his law that planets sweep out equal areas in equal times and shown to be true by Newton from his laws of motion and gravitation. However, it was a leap of faith that the same conservation law was applicable to a figure skater who uses muscle power to spin and even more to subatomic particles. Feynman argues that one must use the Babylonian method to make these leaps since they could not be deduced axiomatically. As long as we don’t have a complete theory of physics, the axiomatic approach will not be useful.

ruveyn

an intriguing discussion

I get the impression that our species devises a system for describing things when we find that it is necessary to do so. As a primitive human, I can imagine seeing two gazelles drinking out of a river without having the word for "two", but still sensing that it is still more than one. From this, I think it may be accurate to say that we discovered quantity, but the language meant to describe it probably needed some work.

The definitions of "discovery" and "creation" are ambiguous when dealing with methods of communication. If math is a language (as in "language of science"), then it is a means to convey information. A means of communication is "created" in the sense that beings make use of unique encodings they engineer. If "discovery" means something along the lines of "to acknowledge for the first time", where can creation and discovery be separated in this context? Was singing/speaking created or discovered?

I suppose if someone made something up, he could say he created the idea, but saying that it was "discovered" might mean testing it objectively against reality. Anything more permissive would probably mean accepting idealism.

I'm tempted to say math was created because the things it describes were discovered, but then again, were irrational numbers discovered on proving the square root of 2? Under those circumstances, axioms led to people saying "Hey, this has to be true, even though I never observed any physical manifestation of it".

The quantity equal to the square root of two can be physically manifested. It's the ratio of the length of a line connecting two opposite corners of a square to the length of one side of said square. It's just that originally most people assumed that a ratio of any two quantities could be expressed as a ratio of integers. The Greek mathematicians/philosophers were the first to show that this naive assumption was a fallacy, in effect discovering the existence of irrational numbers.

Last edited by marshall on 22 Apr 2012, 1:17 pm, edited 2 times in total.

Thanks for the clarification.

EDIT: Fixed quote.

I'd like to note that after doing a lot of research regarding the supposed foundations of mathematics (i.e. the formal language of first-order predicate logic, basic set theoretic axioms, and the establishment of Peano arithmetic) that a certain vicious circularity is unavoidable. Both the grammatical and the semantic description of first-order logic are defined using set-theoretic notions which themselves can only be proved through axioms which are themselves stated in the language of first-order logic. It's all cool to study but there isn't really any constructive starting point. Equivocating different layers of logic and meta-logic with the naive assumption that they are objectively describing the same thing leads to a lot of weird paradoxes.

http://en.wikipedia.org/wiki/Richard%27s_paradox

http://en.wikipedia.org/wiki/Skolem%27s_paradox

Don't forget Godel!

http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems

In the physical world there are no points, lines or squares. The geometrical square is an abstraction and exists only in our heads.

ruveyn

Good point.

If platonic forms cannot be manifested, the ambiguity I mentioned earlier comes back in. If we cannot witness forms, but can draw an understanding of their relationships and apply them to the physical world, where does creation end and discovery begin, and vice versa?

Nature creates the dots (they have to be discovered). Man connects them with abstract mental dotted lines (the is the creative part).

ruveyn

I guess I'm not so quick to completely throw out Platonism. When you try to get to the nuts and bolts of theoretical physics it becomes more and more difficult to differentiate the mathematical model used to describe the physical world from the physical world itself. Of course any particular model we have may easily be proven wrong experimentally. That doesn't necessarily prove that no exact platonic model of the physical universe exists. It only proves that if an exact model does exist we haven't found it.

The notion of "proof" is a Platonism which everyone must accept. Even if you think that "2 + 4 = 4" means



you still have to believe that something called a "proof" really exists. If you write the same proof on three different whiteboards in different colours, you have to believe that they are the "same" proof, in much the same way that a person might abstract the number "three" by looking at several collections of three objects. So it's not possible to be a true formalist.

I think there might be some connection between Godel's Incompleteness Theorem and so-called "semantic paradoxes". The meta-logical proof of the existence of an unprovable statement requires the ability to "code" the formal logical language in which mathematics statements may be posed. Each logical statement within the formal language can be assigned a unique natural number that is itself defined within the formal language. A meta-logical statement of the type "statement n is provable from the axioms" is not possible to express in the formal language, but its formal analog turns out to be a valid formal statement involving natural numbers. It's possible because the entire formal language can be coded into natural numbers, much like an entire ascii file is coded as a large binary number.

Then you can construct a meta-logical statement whose proof or disproof leads to a meta-language paradox, but whose formal analog is a perfectly sound formal statement regarding natural numbers. The meta-language statement is similar to the liars paradox, i.e. if I say "I am lying" the assumption that I am lying determines that I can't be lying, and the assumption that I'm not lying must mean I am lying. If a formal proof of the numerical analog to the meta-logical statement "this statement is not provable from the axioms" exists, the negation follows as a corollary. Similarly, if the negation can be proven, the truth of the original statement follows. Therefore the formal numerical analog to "this statement is not provable from the axioms" is a so-called Godel hypothesis, i.e. a statement about natural numbers that can be neither proved nor disproved unless the formal foundational axioms of set theory are inconsistent.

Last edited by marshall on 23 Apr 2012, 3:06 pm, edited 4 times in total.

Yea. In Peano arithmetic "2 + 2 = 4" is shorthand for the logical string "+(S(S(0))S(S(0)))=S(S(S(S(0))))". But then what the heck is a logical string in Peano arithmetic? It's ordered list of symbols involving functions "+", "x", and "S", the constant "0", along with parenthesis, equality. and a set of logical connectives. You can formally say that a logical proof is a list of logical strings, starting with the formal axioms, that obeys certain "transformation rules" relating each string to one or more previous strings in the list. But then to define the transformations you still have to do some simple proofs at the "meta" level, i.e. outside the formal language, just to show that everything is well-defined. Any necessary meta-proofs have to be taken on faith on a platonic level.

Mathematics exists without human interaction, we just gave things names. For instance, The Golden Ratio is a proportion that naturally occurs in nature and began occurring long before we noticed it was there. The Fibonacci Sequence is a series of numbers, but is also a naturally occurring pattern in the growth patterns of tree leafs and flower petals. The Four Leaf Clover, in example, is considered lucky because 4, not being part of the Fibonacci Sequence, is rarely found to occur naturally in plant growth. This is the reason we hang picture frames in groups of three or five, subconsciously, our minds are draw to perfect proportions as seen in the Fibonacci Sequence and the Golden Ratio. I believe insight and innovation lead to discoveries such as these and allow us to further evaluate and give names to certain phenomena, but that without us, mathematics would continue to exist. We just wouldn't have a name for it.