Math was discovered, not created

Great Debates
#1

Most mathematicians operate on the assumption that they are studying something that already exists: their theorems are discoveries and not creations in contrast with poems and plays. But how is that possible? It sounds like they are saying that the theorems exist before they are contemplated: if so, where are they?

I’m going to make 2 broad arguments: the first is adapted from the physicist Roger Penrose (2007). The second comes from the philosopher Bertrand Russell’s The Problems of Philosophy (1912), chapters 8 - 9. The ideas and good parts are their’s, the slapdash and inaccurate aspects of the presentation are mine.

Consider fractals, in particular the Mandelbrot set. It’s created by a fairly simple rule. Yet it is incredibly elaborate: in fact its detail is basically infinite, or so I understand. Regardless, it’s quite a sizable thing, too big for a single person to hold in their head in its entirety. The set itself (as opposed to the algorithm behind it) simply can’t be an invention. So it must have been discovered.

Buh wah? If the Mandelbrot set was discovered what the hell is it? It has no mass, no energy. How can we say that it exists? [sup]1[/sup]

It’s real. It just has different properties than physical objects, energy and even thoughts. In such cases, it’s useful to introduce new terminology, new definitions, so that we can wrap our heads around unfamiliar ideas. We’ll say that the Mandelbrot set (or mathematics in general) are examples of Universals.

Let’s think a little about mind and matter: we will see that universals are neither:

So again: that something can be inside something else: that’s real. It certainly isn’t merely a thing that is only in our minds. Rocks and sticks are real as well. So is the relation 2+2=4 -if you put together 2 objects with another 2 objects there must be 4 objects-, although that’s harder to see than something being inside something else. The notion of being closer to one object vs another - say a given bird is closer to New York than to Chicago - that universal is real as well. As is the bird.

To pull out some more terminology, we can say that rocks, sticks and thoughts exist while universals like math and being inside something else subsist. Things that exist have a location in time and space. Universals though subsist across time. All time, AFAIK. The property of being in something else isn’t dependent upon someone thinking it and it isn’t dependent upon any particular physical example. It’s not thought and it’s not matter: Russell labels such things as universals.

Now at this point Plato voyages into mysticism[sup]2[/sup], which is why I couldn’t take him too seriously. But we don’t have to do that. Material objects are real, but so are universals. Plato implied that universals were more real than physical things, which were mere examples. But (really!) both entities are real and which category you focus on will vary with personal preference. Personally, I’m an empiricist. But from time to time idealists will post on this message board.
[sup]1[/sup] For those keeping score, the material before this point came from Penrose, chapter 1. The material after it comes from Bertrand Russell (1912). Russell adapted it straight from Plato. I studied Plato in college, but this stuff -core concepts really- sailed over my head. Thanks Professor Russell!

[sup]2[/sup]See eg the allegory of the cave.

Previous threads:
2000: Is math truly objective?
2001: Mathematics: Invented or Discovered?
2001: Is mathematics made up?
2004: Epistemologically is mathematics considered more of an invention, or a discovery?

#2

I’d say math was invented to express realities.
The Pythagorean theroem is not a reality, it’s a human invention that helps us understand reality.

#3

The discovery of patterns that already existed.

#4

Define what you mean by math. All principles have always existed. The symbology used to represent them are inventions. So which one are you talking about?

#5

Well, for the time, Plato was quite rational. The worship of number found in some of Plato and offhand Pythagoras’ writings (or at least, found in the content of his what is sometimes labeled a cult), is better than just plain hardcore religion in terms of rationality.

And while the rest of your post is sort’ve interesting, all of mathematics is invented, not discovered. At least if you count the conscious and the subconscious as the same thing. You could I suppose say that we consciously ‘discover’ all new ideas, because they all come from that part of the brain which isn’t strictly in our control. But apart from that, I don’t know how any Mathematical theorem formula or object could be ‘discovered.’

#6

You nailed it. Principles are universals, they have always existed and they are as real as pieces of matter. I didn’t grasp that until I read Russell.

I agree that the particular symbols must be inventions though. The Arabs invented the character “6”, while the Romans used the character combination “VI”. Those were inventions – they could have taken many forms. But the underlying principle -what they represent- subsists (in Russell’s terminology) or has always existed (in yours).

Those posters who believe that mathematics was invented need to explain how a fractal -eg the Mandelbrot set- that can’t be held in a single human mind could possibly be a human invention.

(Note to reader about the Mandelbrot set: Don’t be intimidated by the name. You’ve probably seen it already. Click the wiki link. http://en.wikipedia.org/wiki/Mandelbrot_set Oh yeah that thing.)

#7

This idea has wide application. A machinist doesn’t make a part, he just carefully unwraps the surrounding material to reveal the part that was in there the whole time.

According to Arlo Guthrie, he doesn’t right songs. They just float by, and if he has a pencil handy he writes it down and calls it his. He says Bob Dillon always has a pencil.

#8

Corrected quote:

#9

I don’t understand what you mean by this. I’d say that the particular reality that the Pythagorean theorem helps us understand is the reality that the square on the hypotenuse of a right triangle is the sum of the squares on the other two sides—i.e. the Pythagorean Theorem itself.

ETA I guess that, to me, “reality” entails not only everything that exists but also everything that subsists, in the language of the OP.

#10

Told you guys, stay away from the brown acid. The one time I’m totally right, nobody listens.

#11

Nobody calls it “math” anymore.

Anyway, IMHO it is discovered. But I haven’t worked out the explanation… the state of affairs is ‘as is’.

#12

Mathematics is a language that we use to imperfectly describe natural principles.

#13

Here is a perfectly grammatical proposition in “mathematical language”:

2+2=17

It is not mathematics. If math is simply “a language that we use [imperfectly to] describe natural principles,” then expression of untrue statements should be just as much math as true statements. Just as ‘Al Gore was the President of the United States in 2012’ is English just as much as 'Barack Obama was President of the United States in 2012."

But ‘2+2=17’ is not math (although it is a well-formed formula (wff) in the language of elementary arithmetic). By contrast, '2+2=4" is a wff and also true, which makes it a mathematical theorem.

#14

Pit thread about that, isn’t there?

#15

Measure for Measure, do you think that things do not come into being until someone puts them in communicable form?

Because if you think so, that’s not Platonism, it’s magic thought. Thales created his theorem as much as Copernicus moved the Earth and the Sun, as much as Kepler changed the racetracks of celestial bodies, as much as Newton brought gravity into being.

#16

I can grab a clump of Copernicus’s Earth. The Sun exists in time and space. I can tell you were in time and space these things have their being.

Where does the Pythagorean Theorem exist? Where can I take a look at the Goldbach Conjecture (an unsolved problem in mathematics, so it would be super helpful it we could get a look at it, by the way)?

#17

I’m going to make a broad, subjective statement here, one that could easily be wrong because I don’t know much about the history of philosophy.

From what I do know, Russell seems to be the guy who discovered and understood meta and how it applies to logic. Earlier philosophers didn’t seem to fully grasp the difference between things and their names. The Pythagorean Theorem is a name. It’s expressed with symbols, another form of names. It is describing a principle which exists in nature whether or not any humans exist to discover it.

#18

This thread is not so much about math, as about epistemology; how we form ideas and their relation to reality, and the relationship between truth and validity.

But I don’t think the OP wants to get into all of that, and I’m too tired right now. But I’ll just ask a question: If someone were in sensory deprivation since birth (or before), how would he understand things like “two-ness” or “pi” or “icosahedron” or “vector calculus”? Totally cut off from reality, I doubt that these abstractions could be invented. And the tree-in-the-forest question: Before there was any life in the universe, did Pi have the same value as now? And if math were an invention, why couldn’t we just make Pi = 3?

#19

The relationship between the sides of a right triangle existed long before Pythagoras wrote it down. It existed before there was even human beings.

#20

That’s what I told the feds when they busted down my door! “This meth??? I just discovered it, I didn’t create it!”

Oh… MATH. Oops.