# Math is invented?

1. Aug 29, 2014

### sonicelectron

Hi everyone, I'm new here so, hello. I guess for my first thread, I'd like to ask whoever thinks math is invented, how do they define "invented"?

2. Aug 29, 2014

### Staff: Mentor

Welcome to the PF.

What is your background in math? What year are you in school? Are you familiar with Peano's Axioms?

http://en.wikipedia.org/wiki/Peano_axioms

That's a good place to start to understand the basis of mathematics IMO.

3. Aug 29, 2014

### sonicelectron

I guess you can say I'm pretty rusty. Highest math I know or "used to know" was diff eq and linear algebra.

4. Aug 29, 2014

### zoobyshoe

In this context "invented" means created by humans for humans, but based on principles existing in nature.

The simple act of counting is an invention. Things aren't counted in nature, they aren't assigned a number between zero and whatever upper limit is operative. The whole notion of keeping a tally is a human one.

5. Aug 29, 2014

### sonicelectron

Thanks for the response. Is there a context in which you think "mathematics" leans more towards discovery than a human creation?

6. Aug 29, 2014

### SteamKing

Staff Emeritus
IMO, all knowledge is a human creation.

By using a few axioms or propositions, one can create a mathematical structure, which may or may not mimic nature. The ancient Greeks, in particular Euclid, started with a handful of propositions, postulates and definitions and constructed a geometry which was thought to describe the earth and the measure of all earthly things.

One of Euclid's postulates, the Fifth, or Parallel, Postulate, caused trouble from the start, almost as soon as the ink was dry on Euclid's 'Elements'. Centuries later, mathematicians like Gauss, Bolyai, and Lobachevsky, were able to show that by altering Euclid's Fifth Postulate, whole new geometries could be derived.

http://en.wikipedia.org/wiki/Non-Euclidean_geometry

By showing that whole new mathematical structures could be derived after changing a few key aspects, mathematicians got to thinking in more depth about the nature of mathematics.

7. Aug 29, 2014

### zoobyshoe

Only in the sense you could say Edison "discovered" that sound could be recorded.

8. Aug 29, 2014

### sonicelectron

So basically, correct me if I'm wrong (it's late), what I'm seeing here so far is that math is merely a set a tools and curiosities created by humans.

9. Aug 29, 2014

### zoobyshoe

Well, your opening post asks for responses from people who think it's an invention.

10. Aug 29, 2014

### sonicelectron

I'm just trying to sum up a basic definition based off of the responses so far.

11. Aug 29, 2014

### sonicelectron

I actually should have made the opening post, "Math is invented because _______."

12. Aug 30, 2014

### phinds

I think your attempt to categorize math as discovered or invented is just an exercise in semantics and not particularly helpful. Those words tend to get very fuzzy sometimes.

13. Aug 30, 2014

### Medicol

14. Aug 30, 2014

### micromass

The axioms and definitions are invented, the rest is discovered :tongue:

15. Aug 30, 2014

### phinds

That's a pretty one-sided view of the article you linked to since the first sentence in it is

16. Aug 30, 2014

### jobyts

Mathematics is logic represented in formal language. Obviously, the formal language is man-made. The logic is with any intelligent form, not just with humans. The prey decides to defend or escape based on the number of predators. It must be able to do the "<", ">" logic in some primitive way .

17. Aug 30, 2014

### mal4mac

Basic logic, distinguishing up to (about) five objects, and being able to distinguish "none" and "many", give us some numbers and basic operations. These, and a few other basic abilities, are given to us by evolution, and give us the beginning of mathematics. You need to start with something!

From most animal films I've seen, the prey doesn't usually act based on *number* of predators. One lion will get you running just as fast as two! Maybe an elephant needs to distinguish "number of lions". Above a few it might decide to amble away. The evolution of being able to distinguish instantly between (say) 4 and 6 objects must (surely) have many causes. It might be caused through gathering activities, as much as prey avoidance.

For instance, if you see a number of apple trees to the left, and a similar number to the right, it might be useful to know six trees from four. If there are many trees, I guess the ability to see that one wood is significantly bigger than another wood, is also useful. But we can't see that there are exactly (say) 67 trees, so I guess the ability to distinguish (say) 65 from 67 just wasn't useful in biological evolution.

Last edited: Aug 30, 2014
18. Aug 30, 2014

### zoobyshoe

I don't think it's an exercise in semantics. It's very easy to feel that Nature is built on mathematics, that the fundamental relationship of everything to everything else is mathematical. I believe it was Galileo who concluded (something like): "God is a mathematician!" Newton seemed to agree.

There are plenty of examples of things that would make people think this way. Take the fact that musical overtones automatically take the form of integer multiples of the fundamental.

http://en.wikipedia.org/wiki/Pythagoras

19. Aug 31, 2014

### mal4mac

What does mathematics have to do with feeling?

The Inquisition were famous for torturing inane statements out of Galileo.

Leopold Kronecker: "God made the integers; all else is the work of man".

Here, I think, Kronecker is making ironic use of the "God concept", and simplifying the actuality, somewhat, to counter Galileo with an equally simple sentence. In that sentence, replace "God" by "evolution" and "the integers" by "between 5 and 9 integers, and a few other simple concepts" and this statement appears reasonable.

20. Aug 31, 2014

### zoobyshoe

My post was about what semantics has to do with feeling, not mathematics.

21. Aug 31, 2014

### Medicol

Last edited by a moderator: May 6, 2017
22. Aug 31, 2014

### homeomorphic

I think I'm with phinds on this one. I think this topic is very semantically confusing. Even one person thinking about it on their own is bad enough, but when you have two people or more, each using their own meaning, chaos is bound to be the result. Well, that's philosophy for you. But here's my take on it.

There are certain aspects of math that are very clearly invented, such as the notation and maybe the axioms, but it gets a little wishy-washy beyond that. Invention suggests freedom to choose. If that's what invented means, no, it's not completely invented. But maybe you allow for some restrictions, as with inventing a new gadget, in order to get it to work properly. Discovery is a term that evokes comparisons with scientific discoveries. In spirit, I think it's more like discovery, to my mind.

Think about tic-tac-toe. The rules are made up, but initially, it might not be obvious that the first player can always win or draw if the opponent plays right. You DISCOVER that you can always draw because you didn't know it at first. It doesn't make sense to me to say that we invented the fact that you can always draw. No, we just figured it out. That's the same sense in which we discovery new formulas and theorems in mathematics (and, in fact, you could formulate tic tac toe in terms of mathematical axioms and proof, if you wanted). That's the general idea. We postulate blah blah blah, which is an invention, perhaps abstracted from and inspired by things in nature or in our own thought processes, and then we discover that if blah blah blah, then more blah blah blah follows. So, we invent the axioms, perhaps, we invent the rules of deduction, but theorems are discovered, if you are going to use the terms invented and discovered in a way that makes sense to me, at least.

23. Aug 31, 2014

### mal4mac

Interesting point. Was the wheel discovered or invented? Was 17 discovered or invented? A caveman mathematician might have learned to count up to 16, and made the next step - did he invent 17? If he found a bunch of shells and counted them up to 16 and realised there was one more he might have said, "I'll call that 17, I've just invented 17". But has he really invented 17, or discovered it? But, certainly, 17 isn't "God given"; it doesn't exist in a kind of "Platonic heaven" as some would have it.

24. Aug 31, 2014

### mal4mac

What if the guy who invented tic-tac-toe started out by trying to invent a game in which optimal players would always draw? Subsequent players might discover they can only draw, but this "feature" was actually invented.

I guess that a theist could argue that theorems are invented by God. Atheists suggest that in an infinite multiverse all possible theorems are created. So no human inventor is needed. But then you might say, speaking a bit loosely, that "the universe" invented the theorems. But it took humans to disover them. So they are invented and discovered!

So, I disagree. The universe invented the axioms, rules of deduction, and theorems. But we discovered them.

25. Aug 31, 2014

### ModusPwnd

Sounds good to me. Math is an invention. That doesn't degrade or lessen it in any way. Also, there certainly can be discovery and experimentation with our inventions.