The principles, theorems, rules, whatever, of mathematics—were (are) they created by humans, or did they always exist? I’m curious what people here think. This isn’t about God or faith—it came up from reading something math-related.

“Nothing personal, but I tend to be suspicious of such questions because some believers use them as Trojan horses.”

I agree, and I regret the suspicion, because it severely limits the response I get to the few topics I have started here. Thus the obligatory disclaimer. This has nothing to do with God gave us math, or humans just make shit up.

Consider language. In different languages, not only the vocabulary is different, but often the rules of grammar and syntax—inflected, not inflected, word order important, not important, etc. Also the ideas can be different. “I will email you a link to a video that you will need Quicktime to watch.” Terrible grammar, words that meant nothing to your parents when they were dating, but you know exactly what the sentence means.

Now consider math. There is one sun, two holes in your nose, three leaflets on a shamrock, four legs on a dog, etc, whether we have names for the numbers, or even care. But what about pi? There is only one perfect circle in nature that I can think of, and you can rarely ever see all of it. It can’t be accurately measured. So is pi part of the world, or part of our world?

Mcalpine, this question is one that is a point of contention between different schools in the philosophy of mathematics. Roughly:

1. Mathematical Platonism (A term coined by Paul Bernays in the early 1930s) is the folk belief of the largest number of mathematicians. It’s best known proponent was the logician Kurt GĂ¶del. It is the idea that mathematical objects have real existence in a Platonic reality. Mathematicians do not create them, but discover them in the same way that Columbus discovered America. Mathematical theorems have objective truth values regardless of whether or not any human mind (or any other mind) knows them. The Pythagorean theorem, for example, was true long before humans evolved, it is something eternal.

2. Mathematical Formalism (A term connected with David Hilbert) is the position that mathematical results are human constructions—a mathematical system is a set of symbols together with rules of grammar and initial axioms (i.e., initial strings of symbols that are assumed without question). Any string of symbols that can be derived from the initial axioms by following the rules of grammar is a true theorem. To ask questions about the “real” existence or not of mathematical objects is fruitless.

3. Mathematical Constructivism (or Intuitionism) A mathematical result is true only to the extent that the mathematician constructs the result and can see it in direct intuition in their own mind. A published proof for a theorem is just a recipe that allows others to do the internal construction that will allow they to see the truth of the claimed result. Intuitionists come in various stripes, but in general they do not accept proof by contradiction because that sort of proof doesn’t provide an actual construction. Very few mathematicians are intuitionists.

There is also a new group of postmodern types who claim that mathematical results are culturally specific and that there is no way to know if a mathematical theorem is true beyond the human minds that have framed it.

The way I would think about it is that mathematical objects do have objective existence, but we construct/invent the means (i.e., the notation and so on) that allows us to discover them. Analogy: the moons of Jupiter were there all along, but we did not discover them until Galileo built a telescope and looked.

A set of axioms, or game rules, is arbitrarily chosen, but the results following from that arbitrary choice, are non-arbitrary.
(the set of legitimate “moves”, for example)

Thanks, Burt, that was very informative. Platonian objects was what I had read about.

Jefe, I was thinking about a rainbow. We never see the whole circle, though, unless you have your lawn sprinkler on at just the right time of day. And then there is the halo around the moon or streetlights when the atmospheric conditions are just right. These aren’t exactly optical illusions, but are hard to pin down in a physical way.

Arildno, that’s an interesting analogy. Both games are based on a standard deck of playing cards, which exists outside of the games, and can be used in different ways in other games. The cards themselves don’t seem to care what the rules are.

Interesting thread. I’m no expert on mathematics, but I would bet that something like the Pythagrean Theorum was indeed discovered, not created or invented. However, as far as our “choice ” of a numbering system is concerned it seems that our particular accepted numerals (0 to 9) are probably arbitrary in some sense. Same with our acceptance of the Base 10 decimal system (there are other differently based systems). My question, I remember reading something about the “discovery” of the zero value, that it was this particular discovery that actually made the process of quantification work properly. If zero was a discovery, then perhaps so were all the rest?

Interesting thread. I’m no expert on mathematics, but I would bet that something like the Pythagrean Theorum was indeed discovered, not created or invented. However, as far as our “choice ” of a numbering system is concerned it seems that our particular accepted numerals (0 to 9) are probably arbitrary in some sense. Same with our acceptance of the Base 10 decimal system (there are other differently based systems). My question, I remember reading something about the “discovery” of the zero value, that it was this particular discovery that actually made the process of quantification work properly. If zero was a discovery, then perhaps so were all the rest?
Bob

That’s a good comment Bob, connected to the importance of notation for doing math. The Greeks, for example, couldn’t easily do algebra because they used Greek (and a few Phonician) letters to stand for numbers so they didn’t have them available to use as unknowns. Likewise, it isn’t possible to write abstract equations using Chinese ideograms. So in part (IMHO) there is a continuing interaction between the development of notation and the discoveries that become possible through that. In that sense, the “discovery” of 0 was the development of a symbol that stood for emptiness, or the addition of nothing, and the adaptation of the number system to include this. That also means that there are (at least) two different forms of mathematical intuition. The intuitive understanding of the notation and the rules for combining symbols together into other strings of symbols, and the intuition that allows going beyond the notation to an internal mental grasping of the mathematical reality beyond the notation. Mathematical Platonists use the first sort of intuition to reach the second, Formalists stick with the first while Intuitionists only accept a limited form of the second. Much of the debate that goes on can be taken in terms of the question of whether or not it is possible to accept the reality of mathematical objects that are not available to the second form of intuition in the Intuitionist sense. That is, we can all grasp the Pythagorean theorem both in terms of its symbolic proof, and also as an immediate internal perception (because we can view the internal construction). So it makes sense to assume that there is some sort of eternal reality to that theorem. But what about a theorem that can be proved symbolically (so, is valid according to the first sort of intuition) but is so complex that we can’t really grasp it in the mind—one of the questions in fact is whether or not there are mathematical truths that we cannot fully grasp.

Thanks, Burt, that was very informative. Platonian objects was what I had read about.

Jefe, I was thinking about a rainbow. We never see the whole circle, though, unless you have your lawn sprinkler on at just the right time of day. And then there is the halo around the moon or streetlights when the atmospheric conditions are just right. These aren’t exactly optical illusions, but are hard to pin down in a physical way.

Arildno, that’s an interesting analogy. Both games are based on a standard deck of playing cards, which exists outside of the games, and can be used in different ways in other games. The cards themselves don’t seem to care what the rules are.

Neither do the basic “objects” in maths, often called “numbers”, “sets” or “structures”.

Other axioms define other number systems you can play with.

I think the “discoveries” of the laws of math are similar to the “discoveries” of the laws of physics; better and better approximations of the physical world. Advances in physics usually come in tandem with advances in mathematics.

There’s an interesting paper which discusses the “Unreasonable Effectiveness” of math applied the the physical world…

So I would venture that, yes, discoveries in math are of pre-existing truths, because beings in another galaxy would most likely come up with similar values (even if expressed in a different number system) for the important constants (e, pi, phi, etc) and the relationships among them.

I think the “discoveries” of the laws of math are similar to the “discoveries” of the laws of physics; better and better approximations of the physical world. Advances in physics usually come in tandem with advances in mathematics.

There’s an interesting paper which discusses the “Unreasonable Effectiveness” of math applied the the physical world…

So I would venture that, yes, discoveries in math are of pre-existing truths, because beings in another galaxy would most likely come up with similar values (even if expressed in a different number system) for the important constants (e, pi, phi, etc) and the relationships among them.

There is a difference between math and physics. In math there are not really things like physical “laws.” Rather, there are rules for combination of symbols according to pre-assumed axioms, things that have been proved, and all the rest. In physics, the so called “laws” are just statements of the best description we currently have of how the world operates. Take the “law of gravity.” Is it the Aristotelian one; the Newtonian one; the Einsteinian one? Or something else? None of these descriptions of gravity is absolute in the same way as the Pythagorean theorem, and no physical law can ever be.

Was mathematics made up by people, incorporated into culture and society and then handed down through the generations? Ideas are memes. Is not mathematics an idea? It is an invented explanation and intrepetation of the abstract, no? 2 plus 2 equals 4. Only if we say so and believe it and tell our kids it is so.

Was mathematics made up by people, incorporated into culture and society and then handed down through the generations? Ideas are memes. Is not mathematics an idea?

Whatever that means.

It is an invented explanation

Eeh?

and intrepetation of the abstract, no?

Eeh?

2 plus 2 equals 4. Only if we say so and believe it and tell our kids it is so.

Nope.
2+2=4 is provable on basis of the chosen definitions of the symbols 2,+,=,4 and the chosen axioms for arithmetic.
The deduction is entirely valid.