Plato's Forms And Mathematics

Metaphysics and Epistemology
1

Here is my main question:

  1. Was mathematics discovered or invented? (Some things to think about, was the pathagoreom theory discovered or invented? What about other mathematic forumalations?)

sub-question:

  1. If math was invented, how did we come about mathematical theoroms? How do we know that a2+b2=c2.

The rest of my post sort of serves as a context for the two questions, but isn’t too important–so feel free not to read it.

During my a discussion in my ancient philosophy class, I said that concepts weren’t created by the subject w/o reality, and that they simply coudn’t exist solely in the object. . .to which another student replied saying that 2+2 will always equal four, regardless if humans existed or not.

So I said that 2+2 only equals 4 because we said that when we saw two similar objects and combined them with two more similar objects, we said that the result is 4. . .but we could have said the result is five. 2 and 4 are merely names we ascribe to the objects being counted.

I don’t think math was *discovered*–it’s not like we lifted up some rock and there in the sand was written: 2+2=4.

If math is the science of measurement, then there must be osmething to measure before math exists. . .but I still don’t see how this relates to math theories and perfect circles/sqs/triangles and what not.

(ps I am pretty ignorant with math, so if you respond in hopes of making sense to me, please keep it layman-like, thanks!)

2

I can help with #1.

Mathmatics is a language used to describe the univese and all of the objects within it. As such it was plainly invented, and it required (and continues to require) a great amount of intellect to create. Languages are clearly inventions rather than discoveries.

3

Here is my main question:

  1. Was mathematics discovered or invented?

It depends on how you want to define the terms ‘discovered’ and ‘invented’. Given a set of axioms and inference rules, all of the theorems which follow from them are, in a sense, immediately determined (ie we could program a computer to succesive churn them out). But this says no more than the fact that given some natural language with a finite number of words (such as English), all possibly 400 pages books are determined in advance (there are only finitely many different sequences of words which can fill 400 pages). But noone would say that a particular novel was discovered rather than invented, even though it was, in a sense, already present in the ‘state space’ of possible English language books.

Having said that, there are certain things in mathematics where the word ‘discover’ seems to make more sense than ‘invent’ - the Mandelbrot set (which was found by mistake) would be a good example, along with a lot of complex analysis, where making a single addition to the real numbers managed to produce a truly amazing amount of theory which noone could possibly have predicted in advance. The imaginary number ‘i’ was ‘invented’ for the purposes of solving certain equations, but introducing it had consequences which went far far beyond this.

  1. If math was invented, how did we come about mathematical theoroms? How do we know that a2+b2=c2.

I’m not sure what youre asking here. If you want to know how a particular theorem was discovered/invented, then read a history of mathematics book that describes the details. But I suspect this isnt what you mean.

During my a discussion in my ancient philosophy class, I said that concepts weren’t created by the subject w/o reality, and that they simply coudn’t exist solely in the object. . .to which another student replied saying that 2+2 will always equal four, regardless if humans existed or not.

I think that this is a bad example, but when it comes to mathematical objects like the Mandelbrot set, we can say they would exist without humans in the sense that alien mathematicians on a planet millions of lightyears away would be able to find the same object. Once you have defined the rules for complex numbers, its always ‘there’.

So I said that 2+2 only equals 4 because we said that when we saw two similar objects and combined them with two more similar objects, we said that the result is 4. . .but we could have said the result is five.

I think this is generally correct. There are infinitely many ways of defining the addition function ‘+’ on the integers, so the choice of which one to use is giong to be determined by our pratical needs. Defining ‘+’ so we get that 2+2=4 is useful in a world where most natural objects obey this law, but if the world were other than what it is, a different definition might be more appropriate. Geomtery is a good analogy here - there are many different possibly geometries, but we use Euclidean in day to day life since thats the one which is best suited to describing the world around us.

I don’t think math was *discovered*–it’s not like we lifted up some rock and there in the sand was written: 2+2=4.

The reason people want to say math was discovered is because of the tendency it has to produce surprises, and ‘link up’ in unexpected ways. There are some things in maths which are truly amazing, and saying that they were ‘invented’ and that the deep structure is just ‘coincidence’ seems inadquate. Focusing on examples like 2+2 = 4 is misleading beacuse theres nothing deep or interesting going on there - you really have to consider more advanced topics to get a feel for the sort of thing that makes mathematicians tend towards Platonism.

4

Was mathematics discovered or invented?

Invented. It’s a method, and methods have to be invented (they aren’t lying around on the ground or under rocks).

  1. If math was invented, how did we come about mathematical theoroms? How do we know that a2+b2=c2.

It depends on exactly what question you wanted to ask, but one answer would be that the Babylonians invented math about 4,000 years ago, and we don’t have enough of their math journals to know the exact historical development of the ideas. Babylonian “Pythagorean triples” have been argued to be the first number-theoretic documents. {Alternatively, Egyptian documents might have been the earliest – that’s an issue for professional archaeologists to debate}. There is sufficient evidence of basic mathematical knowledge (for example ratios) in prehistoric civilizations that we have to assume that some kind of system existed 1,000 years or more before the Babylonians. It is said that the Pythagorean theory was first proven by the Indian mathematician Baudhayana around 800 BJ, but you’d have to read it to see whether it is a “proof” in the sense that we understand it.

If you mean, how did we discover the various relations like the Pythagorean theory or the mathematics of pi, it must have been experimental.

So I said that 2+2 only equals 4 because we said that when we saw two similar objects and combined them with two more similar objects, we said that the result is 4. . .but we could have said the result is five.

More or less. The name of the concept differs dramatically according to what language you’re using; even the principles for forming complex numeric expressions differ significantly. The shapes of the letters we use for representing numbers differ (Greek, Roman, Indic, Arabic, Modern Latin – they say we use “Hindu-Arabic” numerals, but you can’t read either Arabic or Sanskrit numbers without learning it as a new script),

5

Invented. It’s a method, and methods have to be invented (they aren’t lying around on the ground or under rocks).

Not really, its more a body of theorems and structure. The ‘method’ of mathematics has changed a lot over the last 3000 years, so it cant really be considered defining.

More or less. The name of the concept differs dramatically according to what language you’re using; even the principles for forming complex numeric expressions differ significantly. The shapes of the letters we use for representing numbers differ (Greek, Roman, Indic, Arabic, Modern Latin – they say we use “Hindu-Arabic” numerals, but you can’t read either Arabic or Sanskrit numbers without learning it as a new script),

Numerals arent numbers - a sign isnt the thing it signifies. XV, 15 and binary 1111 are all symbols for the same number, and X+V = XV, 10 + 5 = 15, and 1010 + 0101 = 1111 all represent the same equation. But this is saying something different from 2 + 2 = 5. Its not just that we’re going to write the sign ‘5’ instead of ‘4’, its that we are definiting the addition (‘+’) operator differently. Its more comparable to saying that 2 + 2 = 1 (in base 3 arithmetic)

6

Its more comparable to saying that 2 + 2 = 1 (in base 3 arithmetic)

11, surely.

7

11, surely.

Sorry, I meant in base 3 modular arithmetic.

8

The ‘method’ of mathematics has changed a lot over the last 3000 years, so it cant really be considered defining.

Okay, it would have been better to say that it’s a meta-method. At any stage it is a particular method. But I take it you only consider the sentences generated by mathematicians to be mathematics.

Numerals arent numbers - a sign isnt the thing it signifies.

The thing that it signifies is a concept (which cannot exist without a conceptual consciousness). We can’t possibly be talking about concepts, because that was what this other guy in class claimed was specifically irrelevant. Hence we only have the non-conceptual stuff, the expressed labels. We could still say that 2+2=5. The numerals are 1, 2, 3, 5, 6, 4, 7, 8, 9… pronounced shuj, er, three, quatro, panca, rokyo, sab`a, åtte, kenda.

9

But I take it you only consider the sentences generated by mathematicians to be mathematics

I dont think theres a meaningful answer to this question. We genearlly apply the label ‘mathematician’ to anyone who does mathematics, so it would be true by definition that only sentences generated by mathematicans would be mathematics. Although I suppose an advanced artificial intelligence program can generate new non-trivial theorems.

The thing that it signifies is a concept (which cannot exist without a conceptual consciousness).

Well, its not inherant in the concept ‘2’ that 2+2=4, since there are infinite ways of defining relations on the real numbers. If we were taking the combination of water droplets as our paradigm case, we might want to say that n + n = 1 for all integers n>0, but this wouldnt be very useful.

We can’t possibly be talking about concepts, because that was what this other guy in class claimed was specifically irrelevant. Hence we only have the non-conceptual stuff, the expressed labels. We could still say that 2+2=5. The numerals are 1, 2, 3, 5, 6, 4, 7, 8, 9

Yeah, this is one way of saying that 2+2=5. But I dont think it was what he meant. Youre using the same numbers and the same function, youre just writing them differently.

2+2=5 might be a statement we’d consider correct if we lived in a world where whenever you combined 2 pairs of objects, a fifth one spontaneously sprang into existence. But since this does not describe the reality we live in, 2+2=4 is more sensible.

10

Well, its not inherant in the concept ‘2’ that 2+2=4, since there are infinite ways of defining relations on the real numbers.

But it is an inescapable consequence of the concepts “2”, “4”, “=” and “+”. The concepts, not the symbols. You have to be careful, of course, in not thinking that the concept “+” is the same concept as “putting together”.

Yeah, this is one way of saying that 2+2=5. But I dont think it was what he meant. Youre using the same numbers and the same function, youre just writing them differently.

He who? The point is that either you are taking about this stuff conceptually or you are talking about it in terms of graphic symbols. You cannot say that “2+2=4” is a universal truth that would be true if man had never existed. All concepts are man-made: numbers do not exist without man. Certain sound or shapes might exist without man, like “1” or [tu] – the numerals – but numbers cannot exist without man. Also, concepts like “addition” and “comparison” require man.

11

Sorry, I meant in base 3 modular arithmetic.

Ah, gotcha.

12

Was mathematics discovered or invented?

Discovered, as all scientific-ish things are.

So I said that 2+2 only equals 4 because we said that when we saw two similar objects and combined them with two more similar objects, we said that the result is 4. . .but we could have said the result is five. 2 and 4 are merely names we ascribe to the objects being counted.

True, but more importantly, the quantity (that is currently referred to as) ‘2’ when added to a similar quantity will always yield the quantity (which we currently refer to as) ‘4.’

13

But it is an inescapable consequence of the concepts “2”, “4”, “=” and “+”. The concepts, not the symbols. You have to be careful, of course, in not thinking that the concept “+” is the same concept as “putting together”.

Actually yeah, youre right. I was making the same mistake I accused you of making (I was confusing the symbol ‘+’ with the addition function).

14

The nature of a concept seems so much more clear to me than the nature of mathematics. Had someone said, “But love will always exist even if humans aren’t around,” it would almost be self-evident that he was wrong.

But in terms of discovery, I don’t think any sciences are discovered–at least not like man has discovered natural laws, such as gravity. So in other words, science (or mathematics) is invented to discover.

Not really, its more a body of theorems and structure. The ‘method’ of mathematics has changed a lot over the last 3000 years, so it cant really be considered defining.

I’m not sure I understood this. Would you elaborate?

Also, in terms of “perfect triangles and perfect circles,” how do we come about these concepts if they don’t exist in reality–or maybe the question is, How does one come about the concept “perfection?”

If I talk about the perfect basketball shot–it is perfect b/c however it was shot, it made the ball go directly through the hoop. But if I talk about a perfect circle, it is perfect not b/c it accomplishes some goal, but because each point on its circumference is equidistant from the circle’s midpoint. (So it seems perfection is used in two senses here: 1) Something can be considered perfect if it brings about the desired end, and 2) I don’t know. . .why would we say that a perfect circle is one who’s curve is everywhere equidistant from the center–there must be some desired end we are trying to obtain with this definition of perfection, if not, I’m at a loss)

15

The nature of a concept seems so much more clear to me than the nature of mathematics. Had someone said, “But love will always exist even if humans aren’t around,” it would almost be self-evident that he was wrong.

But in terms of discovery, I don’t think any sciences are discovered–at least not like man has discovered natural laws, such as gravity. So in other words, science (or mathematics) is invented to discover.

Are you saying that it is both invention and discovery? I could kinda agree with that, but not with it being invention only (or even invention mostly.)

IMO Observing a scientific fact or recognizing new mathematical knowledge (or even objective philosophy?) is different from inventing a lightbulb or writing fiction (or non-fiction I suppose.)

16

IMO Observing a scientific fact or recognizing new mathematical knowledge (or even objective philosophy?) is different from inventing a lightbulb or writing fiction (or non-fiction I suppose.)

What is the difference that you are thinking of?

17

Well, for starters, scientific and mathematical facts exist regardless of whether anyone knows them or not, whereas inventions and novels do not.

18

Well, for starters, scientific and mathematical facts exist regardless of whether anyone knows them or not, whereas inventions and novels do not.

What’s a mathematical fact?

19

E.g. Something along the lines of

“the three angles of a triangle sum to 180 degrees” or

“the derivative of a function of the form y = Ax^B equals (AB)x^(B-1).”

20

“the three angles of a triangle sum to 180 degrees”

Of course, triangles and degrees don’t exist independent of a conceptual consciousness which creates these ideas. But clearly it isn’t automatic that awareness of the concept “triangle” and “degree” creates awareness of this fact: I think we would have to say that knowledge of this fact is implicit in the other two concepts (plus others, but let’s keep this simple), that is, it is an idea that can be created on the basis of other ideas. So it’s an existing fact in the sense that this relationship between concepts is potentially graspable.

The light bulb could be called a “physical fact”, in that it is possible to grasp aspects of the universe which could lead to the seeing that a light bulb could be created (just as given the basic mathematical concepts, you could see that the 180 degree conclusion is also true).

So I don’t see what the difference is between a mathematical conclusion and an invention. I was thinking that in inventing, you often have to solve some intermediate problem. But that’s analogous to a lemma in mathematics, so I still don’t see what the difference is between mathematical and scientific inventions. Are you just referring to the presumed earth-shatteringness of science vs. technology?