Math discovered or invented?

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tonykng
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Math discovered or invented?

Postby tonykng » Fri May 28, 2010 3:06 am UTC

I was having a debate about this and was wondering what you guys thought. Is math inate to the nature of the universe and as such did man discover it, or is it an abstract concept created by man to describe the universe and so man invented it? I am of the opinion that math is inate and discovered.

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Re: Math discovered or invented?

Postby Govalant » Fri May 28, 2010 3:09 am UTC

The best word I've found is 'revealed'.
Now these points of data make a beautiful line.

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Re: Math discovered or invented?

Postby Shokk » Fri May 28, 2010 3:16 am UTC

I think it's a silly debate either way, it's math, it works well enough when we use it for certain things, look at the universe how you will, it is what it is.
I'd say it's suited for human mental consumption and logical processings, and it's tailor-made by its original tailors as such, so I'd say it was invented to deal with the world. You can make the world fit such-and-such patterns and all very neatly but the perfect mathematical models never translate completely to the real world. The world is always too complex to fit perfectly into the ways we try to explain it.

I've always thought math was very....romantic as far as "fitting reality" goes.
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tonykng
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Re: Math discovered or invented?

Postby tonykng » Fri May 28, 2010 3:18 am UTC

Shokk wrote:I think it's a silly debate either way, it's math, it works well enough when we use it for certain things, look at the universe how you will, it is what it is.

I've always thought math was very....romantic as far as "fitting reality" goes.


Along with my math degree I have a philosophy minor, so I enjoy debating silly things. When you can marry that with Math I'm a happy camper.

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Re: Math discovered or invented?

Postby achan1058 » Fri May 28, 2010 4:24 am UTC

Depends on how you look at it. In one way, it's nothing more than a game of string manipulation...... (axiomatic set theory)

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Re: Math discovered or invented?

Postby ++$_ » Fri May 28, 2010 4:49 am UTC

Obviously mathematics exists independently of humans and so is discovered rather than invented. The Pythagorean theorem, for example, is true whether it has been discovered or not.

However, it's reasonable to use the term "invented" colloquially in order to do justice to the ingenuity that is sometimes required in the mathematical discovery process.

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Re: Math discovered or invented?

Postby TheWaterBear » Fri May 28, 2010 4:57 am UTC

It's our attempt to quantify the world.

Invented.

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Re: Math discovered or invented?

Postby MHD » Fri May 28, 2010 5:25 am UTC

Well, considering that we have certain arbitrary conventions, makes me say invented.
Considering pure logic proving everything except the paradoxes, makes me say discovered.
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Re: Math discovered or invented?

Postby z4lis » Fri May 28, 2010 5:34 am UTC

++$_ wrote:The Pythagorean theorem, for example, is true whether it has been discovered or not.


The truth of the Pythagorean rests upon a few geometrical and logical assumptions that we, as humans, have meshed over what we perceive in reality. The assumptions leading up to the Pythagorean theorem were not handed down from another world. People examined how the geometry around them behaved and crafted a symbolic representation for it. In that symbolic representation, a particular statement can be derived. It's still a man-made thing.

It's almost as if saying that because in my RPG setting, a particular, say, sword exists, that that game item is eternal and independent of human activity. It was there all along, just waiting for creatures to come along and "discover" it.
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Re: Math discovered or invented?

Postby BlackSails » Fri May 28, 2010 5:47 am UTC

Math exists independant of man. If we had never named the sine function, there would still be the same relationship between sides and angles of a triangle. If nobody had bothered to ever count past ten, there would still be groups of 11 things.

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Re: Math discovered or invented?

Postby Argency » Fri May 28, 2010 6:13 am UTC

The assumptions leading up to the Pythagorean theorem were not handed down from another world. People examined how the geometry around them behaved and crafted a symbolic representation for it. In that symbolic representation, a particular statement can be derived. It's still a man-made thing.

It's almost as if saying that because in my RPG setting, a particular, say, sword exists, that that game item is eternal and independent of human activity. It was there all along, just waiting for creatures to come along and "discover" it.


That's not a fair analogy at all. Sure, its true that if your in-game item had never been coded no one could ever wield it, just like if the Pythagorean hadn't been discovered noone could ever use it to calculate. But if the Pythagorean formula had never been written before, and I went to walk diagonally across a 30m x 40m oval, I'd still have to walk 50m. On the other hand, if noone had ever coded your sword, it would not give +3 to badassery, which it presumably does, having been coded.

So, sure, its an entirely arbitrary human assumption that "c" means "the length of the longest side" and that "+" means "plus", etc, but it isn't an arbitrary assumption that if you agree on all those meanings, the statement [math]a^2 + b^2 = c^2[/math] is true. Rather, it's a proven fact about the world, which was true before it was proven.

You might say that whilst mathematical truths aren't handed down to us from another world, they are handed down to us from this world.

EDIT: wow, I said oval here meaning soccer pitch, but it sounds like I meant oval the shape. Not sure yet if anyone's picked me up on it but if they have, sorry. Obviously oval shapes have little to do with the Pythagorean.
Last edited by Argency on Sat May 29, 2010 2:14 am UTC, edited 1 time in total.
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Re: Math discovered or invented?

Postby ++$_ » Fri May 28, 2010 6:22 am UTC

z4lis wrote:It's almost as if saying that because in my RPG setting, a particular, say, sword exists, that that game item is eternal and independent of human activity. It was there all along, just waiting for creatures to come along and "discover" it.
Sure -- why not? It's just not a very important discovery, that's all.

This is because there are two senses in which mathematics is a discovery. The first is that it flows as a priori knowledge from some limited set of axioms (just as your RPG sword does). But secondly, it is also an attempt to describe the world as it appears. In this sense, it is discovered in the same way that gravity is discovered. This is why the Pythagorean theorem feels more like a discovery than an RPG sword -- because it's a discovery about this world, rather than an imaginary one.

There's also another problem here, which is that the word "invention" has two meanings. In one sense, an invention is anything that humans think of and put to use. In that sense, mathematics is an invention, because we definitely thought of it, and we definitely put it to use. But in another sense, something invented is something made up. Mathematics isn't made up.

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Re: Math discovered or invented?

Postby PleasingFungus » Fri May 28, 2010 6:45 am UTC

Argency wrote:
The assumptions leading up to the Pythagorean theorem were not handed down from another world. People examined how the geometry around them behaved and crafted a symbolic representation for it. In that symbolic representation, a particular statement can be derived. It's still a man-made thing.

It's almost as if saying that because in my RPG setting, a particular, say, sword exists, that that game item is eternal and independent of human activity. It was there all along, just waiting for creatures to come along and "discover" it.


That's not a fair analogy at all. Sure, its true that if your in-game item had never been coded no one could ever wield it, just like if the Pythagorean hadn't been discovered noone could ever use it to calculate. But if the Pythagorean formula had never been written before, and I went to walk diagonally across a 30m x 40m oval, I'd still have to walk 50m. On the other hand, if noone had ever coded your sword, it would not give +3 to badassery, which it presumably does, having been coded.

So, sure, its an entirely arbitrary human assumption that "c" means "the length of the longest side" and that "+" means "plus", etc, but it isn't an arbitrary assumption that if you agree on all those meanings, the statement [math]a^2 + b^2 = c^2[/math] is true. Rather, it's a proven fact about the world, which was true before it was proven.

You might say that whilst mathematical truths aren't handed down to us from another world, they are handed down to us from this world.

But the thing is, that's not true. Space is non-Euclidean. For most cases, it's very nearly Euclidean (unless you're wandering around near black holes or other supermassive objects); but, in a strict mathematical sense, [imath]a^2 + b^2 = c^2[/imath] does not hold. It's just a very useful approximation.

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Re: Math discovered or invented?

Postby bitsplit » Fri May 28, 2010 12:47 pm UTC

The truth is that there are two aspects to mathematics, one discovered, and one invented. The quintessential truth inherent to reality which lies at the core of mathematics is discovered. The methods, symbols, and manipulations of those symbols both physical and abstract which are agreed upon by convention by the human race in order to convey, and discover those quintessential truths is invented.

But the thing is, that's not true. Space is non-Euclidean. For most cases, it's very nearly Euclidean (unless you're wandering around near black holes or other supermassive objects); but, in a strict mathematical sense, [imath]a^2+b^2=c^2[/imath] does not hold. It's just a very useful approximation.


Yes, but does the inherent truth that space is non-euclidean require you to invent it to be true? Will the planets and stars move in a different way because you conceived of non-euclidean geometry? No. Do we understand geometry better now as a human race because of the advances of the study of mathematics? Yes. Also, there is a difference between Physics and Mathematics. Physics and Geography inspired the Pythagorean Theorem, that is true. That it has been recently shown to be an imperfect method of measuring distance doesn't make it any less true mathematically.

There are mathematical truths and falsehoods that cannot be proven. Does that make them any less true or false? We cannot invent truth or falsehood, it simply is. We invent the way to discover it.
Last edited by bitsplit on Fri May 28, 2010 1:21 pm UTC, edited 1 time in total.

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Re: Math discovered or invented?

Postby majikthise » Fri May 28, 2010 1:08 pm UTC

I'd really really like to able to justify being a mathematical Platonist- it seems so elegant, and suggests a portrayal of mathematicians as explorers toiling on a noble quest to discover hidden knowledge and other warm fuzzy stuff like that. The alternative of picking and choosing which formal theories to use (and working out which things work- and how- inside each one) is somehow less satisfying; but on the up side you gain greater diversity, and studying the plethora of relative consistency and independence results comprises a fascinating field in and of itself.
(It also makes it more difficult to pour scorn on the other sciences for exclusively dealing with base subjects such as the "real world", instead of investigating eternal truths- although it's a matter of opinion whether this is good or bad!)
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Re: Math discovered or invented?

Postby Tirian » Fri May 28, 2010 1:34 pm UTC

majikthise wrote:I'd really really like to able to justify being a mathematical Platonist- it seems so elegant, and suggests a portrayal of mathematicians as explorers toiling on a noble quest to discover hidden knowledge and other warm fuzzy stuff like that.


I don't think that formalism is any less romantic. It casts us as engineers, trying to invent and perfect tools from scratch that allow us to survive and thrive in our environment.

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Re: Math discovered or invented?

Postby majikthise » Fri May 28, 2010 1:49 pm UTC

Tirian wrote:
majikthise wrote:I'd really really like to able to justify being a mathematical Platonist- it seems so elegant, and suggests a portrayal of mathematicians as explorers toiling on a noble quest to discover hidden knowledge and other warm fuzzy stuff like that.


I don't think that formalism is any less romantic. It casts us as engineers, trying to invent and perfect tools from scratch that allow us to survive and thrive in our environment.

Personally I don't see how any portrayal of us as engineers can possibly be as romantic... haha.
Honestly I can't possibly conceive completely agreeing with one side or the other- the heart says Platonism but the head says pluralism/formalism. The only thing I'm certain about is disagreeing with the Luddites! (Sorry, Constructivists :wink: )
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Re: Math discovered or invented?

Postby Argency » Fri May 28, 2010 2:32 pm UTC

Pleasing Fungus
But the thing is, that's not true. Space is non-Euclidean. For most cases, it's very nearly Euclidean (unless you're wandering around near black holes or other supermassive objects); but, in a strict mathematical sense, [imath]a^2 + b^2 = c^2[/imath] does not hold. It's just a very useful approximation.


Okok. Fair point. But I think that's irrelevant to my argument, really. It's still true that if I walked diagonally across a 30m x 40m oval in Euclidean space and on a flat surface then I would have to walk 50m, whether or not someone had come up with the Pythagorean. So the truth of the Pythagorean theorum is a fact about one type of space (whether or not its the space we live in), independent from any human affairs.

Spoiler:
Besides, its only true to say that space is generally non-Euclidean. There are places where space has negative curvature and places where it has positive curvature, and since space is continuous this means that there must also be boundary regions (even if they're infintesimally narrow) where space is completely flat, and where Euclidean geometry therefore holds.
Spoiler:
Finally, the Pythagorean was only an example, and it wasn't my example. Using non-Euclidean geometry you can make true mathematical statements about walking across ovals that hold in the real world, and which would hold even if there weren't mathematicians around to make them.
Spoilered for brevity, and because I'm new to spoiler tags and I like playing with them.
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Re: Math discovered or invented?

Postby BlackSails » Fri May 28, 2010 2:38 pm UTC

PleasingFungus wrote:
You might say that whilst mathematical truths aren't handed down to us from another world, they are handed down to us from this world.

But the thing is, that's not true. Space is non-Euclidean. For most cases, it's very nearly Euclidean (unless you're wandering around near black holes or other supermassive objects); but, in a strict mathematical sense, [imath]a^2 + b^2 = c^2[/imath] does not hold. It's just a very useful approximation.


The pythagorean theorem is for euclidian space, I didnt think that really needs to be stated. For space that isnt flat, then you get a generalized pythagorean theorem.

Furthermore, space itself has curvature regardless of if we have written down axioms of differential geometry or not.

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Re: Math discovered or invented?

Postby Tirian » Fri May 28, 2010 4:12 pm UTC

majikthise wrote:
Tirian wrote:
majikthise wrote:I'd really really like to able to justify being a mathematical Platonist- it seems so elegant, and suggests a portrayal of mathematicians as explorers toiling on a noble quest to discover hidden knowledge and other warm fuzzy stuff like that.


I don't think that formalism is any less romantic. It casts us as engineers, trying to invent and perfect tools from scratch that allow us to survive and thrive in our environment.

Personally I don't see how any portrayal of us as engineers can possibly be as romantic... haha.
Honestly I can't possibly conceive completely agreeing with one side or the other- the heart says Platonism but the head says pluralism/formalism. The only thing I'm certain about is disagreeing with the Luddites! (Sorry, Constructivists :wink: )


My image is of the Professor from Gilligan's Island making complex devices primarily out of coconut shells because that's all he had. To me, that Robinson Crusoe style of innovation is as compelling as the explorers of yore.

To me, it's like the quip around here about the Axiom of Choice: Platonism is clearly both true and false. Imagine a society of non-humans whose intelligence is comparable to our own (be in dolphins or aliens or what have you). I can't conceive that they never had a need to count, and after a few eons of fumbling like we did they'd have come across something that we know must be isomorphic to Peano arithmetic. But would "dolphin calculus" look essentially like ours? I lean against it; I think our calculus is tied up profoundly with the history of Newton and the problems he was trying to solve, and even if dolphins got together and tried to come up with a model for hydrodynamics it would make it easy to solve the problems that they had and not the problems we had. Of course, a Platonist could argue even then that we're both peering into some mystical Truth, but that's different from saying that the specific models that we use were waiting for us to discover them.

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Re: Math discovered or invented?

Postby z4lis » Fri May 28, 2010 4:31 pm UTC

Argency wrote: But if the Pythagorean formula had never been written before, and I went to walk diagonally across a 30m x 40m oval, I'd still have to walk 50m.


Ovals and meters? Those are human abstractions! Sure, walking around might demonstrate that something like the Pythagorean theorem would be a reasonable thing to believe, but that is not a proof! You're using real-world analogies and arguments to give a rigorous argument for the truth of something. Isn't that about like doing analysis by drawing a few pictures? Sure, it gives powerful understanding, but a proof requires a series of formally derived statements.

What about a natural language, perhaps English? English can describe the world, and often in a far more concise and simple way than mathematics does. Does that mean that the language was "discovered" by man to be used? What properties of mathematics make it seem as if what you're doing is a timeless, indestructible thing that was, is, and will be for all time? Is it because you feel the things we name would exist independently of our naming them?

I'd really really like to able to justify being a mathematical Platonist- it seems so elegant, and suggests a portrayal of mathematicians as explorers toiling on a noble quest to discover hidden knowledge and other warm fuzzy stuff like that.


I used to be the same way. I've since then dropped that viewpoint (as is evident) and I've found that the subject is no less enjoyable. Theorems and proofs are just as beautiful, clever, and elegant, but only in the same way I find a poem or building beautiful, clever, or elegant.

Someone above mentioned something like "there would be groups of 11 objects regardless of whether or not anyone bothered to count that high". Of course there would be. But would the concept of a number be around? Where do I find Graham's number? If numbers are "real", then this existing is a direct implication of having the concept of "eleven" along with the bundle of natural numbers. We take an association between groups, in this case the idea of quantity, and we can find collections of objects that fall under some of those association groups. But then we, the curious little bastards we are, use our own noggins to extend and polish that idea beyond anything reality has to offer. Is that really such an unromantic, awful way to look at mathematics?
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Re: Math discovered or invented?

Postby bitsplit » Fri May 28, 2010 5:02 pm UTC

I believe the main argument remains tied to a fallacy: we refer to mathematics to refer to two different things, the truths that are proven, and the human devices used to obtain knowledge of those truths. Most people who argue mathematics are invented ally with the second object, whereas those who believe it is discovered ally with the first. As people capable of abstraction, I would assume we are capable of disambiguating between the two.

Maybe someone disagrees with me, but I find it similar to an argument about whether fruits are sweet or sour, when one person is talking about grapes, and the other about lemons.

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Re: Math discovered or invented?

Postby Shokk » Fri May 28, 2010 5:17 pm UTC

BlackSails wrote:Math exists independant of man. If we had never named the sine function, there would still be the same relationship between sides and angles of a triangle. If nobody had bothered to ever count past ten, there would still be groups of 11 things.

I'd say it exists highly dependent on man's mind and his faculties.
WE came up with the notion of 3-sided shapes on perfect planes, the sine function is like saying "and thus, Zeus begat Athena, and she was smart and stuff." at least as far as being relevant to reality goes.
IT was we who decided to start counting things and grouping them by...what? Weight? Similarity? number of hairs? color? It's arbitrary.
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Re: Math discovered or invented?

Postby BlackSails » Fri May 28, 2010 6:03 pm UTC

z4lis wrote:
Argency wrote: But if the Pythagorean formula had never been written before, and I went to walk diagonally across a 30m x 40m oval, I'd still have to walk 50m.


Ovals and meters? Those are human abstractions! Sure, walking around might demonstrate that something like the Pythagorean theorem would be a reasonable thing to believe, but that is not a proof! You're using real-world analogies and arguments to give a rigorous argument for the truth of something. Isn't that about like doing analysis by drawing a few pictures? Sure, it gives powerful understanding, but a proof requires a series of formally derived statements.


His point was that it takes the same distance regardless of what you have proven.

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Re: Math discovered or invented?

Postby GyRo567 » Fri May 28, 2010 7:25 pm UTC

Just to quickly clarify two things that have been mistakenly assumed:

Mathematics has no objective existence.

Mathematics has nothing in particular to do with this world, nor with any other. There is no particular logic or math out of which the world is made.
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Re: Math discovered or invented?

Postby BlackSails » Fri May 28, 2010 8:35 pm UTC

GyRo567 wrote:Just to quickly clarify two things that have been mistakenly assumed:

Mathematics has no objective existence.

Mathematics has nothing in particular to do with this world, nor with any other. There is no particular logic or math out of which the world is made.


The world is made entirely of math.

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Re: Math discovered or invented?

Postby squareroot1 » Fri May 28, 2010 9:47 pm UTC

Oi, who made this mess?

How about everyone take a couple deep breaths, and instead of posting, just go read a book? Maybe Anathem by Stephenson?

GyRo567 wrote:Mathematics has no objective existence.

Mathematics has nothing in particular to do with this world, nor with any other. There is no particular logic or math out of which the world is made.

Mathematics has an objective existence.

Mathematics has everything in particular to do with this world, as with any other. There is some particular logic or math out of which the world is made.

I can make bold statements without presenting any supporting evidence too. We both deserve big fat "Citation Needed"s.

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Re: Math discovered or invented?

Postby Blatm » Fri May 28, 2010 11:42 pm UTC

BlackSails wrote:
GyRo567 wrote:Just to quickly clarify two things that have been mistakenly assumed:

Mathematics has no objective existence.

Mathematics has nothing in particular to do with this world, nor with any other. There is no particular logic or math out of which the world is made.


The world is made entirely of math.


The world is made entirely of math.


More seriously, I think bitsplit hit the nail on the head:

bitsplit wrote:I believe the main argument remains tied to a fallacy: we refer to mathematics to refer to two different things, the truths that are proven, and the human devices used to obtain knowledge of those truths. Most people who argue mathematics are invented ally with the second object, whereas those who believe it is discovered ally with the first. As people capable of abstraction, I would assume we are capable of disambiguating between the two.

Maybe someone disagrees with me, but I find it similar to an argument about whether fruits are sweet or sour, when one person is talking about grapes, and the other about lemons.

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Re: Math discovered or invented?

Postby Argency » Sat May 29, 2010 2:34 am UTC

z4lis wrote:
Spoiler:
Ovals and meters? Those are human abstractions! Sure, walking around might demonstrate that something like the Pythagorean theorem would be a reasonable thing to believe, but that is not a proof! You're using real-world analogies and arguments to give a rigorous argument for the truth of something. Isn't that about like doing analysis by drawing a few pictures? Sure, it gives powerful understanding, but a proof requires a series of formally derived statements.

What about a natural language, perhaps English? English can describe the world, and often in a far more concise and simple way than mathematics does. Does that mean that the language was "discovered" by man to be used? What properties of mathematics make it seem as if what you're doing is a timeless, indestructible thing that was, is, and will be for all time? Is it because you feel the things we name would exist independently of our naming them?

I agree with bitsplit completely, we're arguing about two different things. You're talking about mathematical language and I'm talking about the truths we use that language to express. I think its disingenuous to interpret my arguments as though I was talking about mathematical language though: obviously I'm not trying to say that our mathematical language is writ in the sky, that would be a stupid argument. What I'm trying to say is that the truths which we determine through mathematical language are inherently true, and would be true even if we never figured them out.

Yes, the things we name would exist independently of our naming them. That's exactly my argument and you managed to summarise it very elegantly, before going on to misrepresent it completely. Are you really trying to argue their existence is dependent on our naming them? Sure, English is a construction, but when I talk about trees and rocks I'm talking about something real which I have discovered in the world and then come up with a word for. Same thing with triangles and the pythagorean.
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Re: Math discovered or invented?

Postby Cleverbeans » Sat May 29, 2010 4:08 am UTC

Argency wrote:Are you really trying to argue their existence is dependent on our naming them? Sure, English is a construction, but when I talk about trees and rocks I'm talking about something real which I have discovered in the world and then come up with a word for. Same thing with triangles and the pythagorean.


Except that triangles don't exist in the real world, they're merely simplifications of nature we're capable of articulating. Mathematical objects are in general defined, their dependency on language should be obvious from that alone. I'd say that mathematical objects are more like unicorns, they bear enough of a resemblance to horse to be practical (I can ride a unicorn, therefore I can ride a horse, etc.) but it's too far reaching to say they exist.
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Re: Math discovered or invented?

Postby BlackSails » Sat May 29, 2010 5:39 am UTC

There are actual instatiated mathematical objects though, depending on your philosophical view of quantum. Take the wavefunction. It is nothing more than a mathematical abstraction, but it does exist.

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Re: Math discovered or invented?

Postby afarnen » Sat May 29, 2010 5:50 am UTC

Cleverbeans wrote:Except that triangles don't exist in the real world, they're merely simplifications of nature we're capable of articulating. Mathematical objects are in general defined, their dependency on language should be obvious from that alone. I'd say that mathematical objects are more like unicorns, they bear enough of a resemblance to horse to be practical (I can ride a unicorn, therefore I can ride a horse, etc.) but it's too far reaching to say they exist.


I think you're confusing two separate meanings of "exist": existing in the real world, and existing mathematically.

Horses exist in the sense that they can be objectively (more or less) perceived by the five senses in this world, no matter who you are.

Triangles "exist" in the sense that they can be objectively reasoned about, independent of what notation you use to do it.

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Re: Math discovered or invented?

Postby Argency » Sat May 29, 2010 6:36 am UTC

Cleverbeans wrote:
Spoiler:
Except that triangles don't exist in the real world, they're merely simplifications of nature we're capable of articulating. Mathematical objects are in general defined, their dependency on language should be obvious from that alone. I'd say that mathematical objects are more like unicorns, they bear enough of a resemblance to horse to be practical (I can ride a unicorn, therefore I can ride a horse, etc.) but it's too far reaching to say they exist.

When you say "I can ride a horse", you're not talking about any particular horse that exists. If you were, you'd have specified which one. What you're talking about is the ideal of a horse - the bit of a horse you'd have left if you took away all its individual quirks (like having a white spot on its nose or being called Pharlap, etc). I agree that that "ideal" of horsey-ness exists, and that "being rideable" is one aspect of it. Its also true that there exist tangible instantiations of horsey-ness - each and every horse in the world consitutes one. So its true to say "horses exist" (because you can touch them) but its also true to say "a horse can be ridden", even though you're only talking about the idea of a horse, not any specific, real horse. It's also true that unicorns are rideable (because that's part of being a unicorn), but it isn't true that unicorns exist.

Similarly, even if triangles weren't instantiated in the real world, it would still be true that [math]a^2 + b^2 = c^2[/math] because that's an inherent part of trianglehood, not something that relies on the specific properties of instantiated triangles. I don't agree that triangles aren't instantiated, though, because any three points in space that don't lie on the one line can be joined by three, forming a triangle. Triangles don't have to be made out of anything because there's no building material specified by the concept of trianglehood, so it doesn't matter if there's nothing actually occupying those points in space: its still true that the pythagorean holds for any three points which constitute a right-angled triangle. Even if humans weren't around to think about triangles all this would be true.
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Re: Math discovered or invented?

Postby GyRo567 » Sat May 29, 2010 6:50 am UTC

Argency wrote:Sure, English is a construction, but when I talk about trees and rocks I'm talking about something real which I have discovered in the world and then come up with a word for. Same thing with triangles and the pythagorean.

The difference is that 'that tree' and 'that rock' denote things with objective existence. (In the sense of the material world) There is no such thing as a triangle in our universe. It is pure abstraction. Mathematics involves relations of ideas, not matters of fact about our particular world. If it were otherwise, we would have another empirical science, not a rational system. It would not be a purely deductive endeavor.
(Edited for pedantry, though I still don't know which tree 'that tree' is.)

squareroot1 wrote:Mathematics has an objective existence.

Mathematics has everything in particular to do with this world, as with any other. There is some particular logic or math out of which the world is made.

I can make bold statements without presenting any supporting evidence too. We both deserve big fat "Citation Needed"s.

I'll be brief on the first point: A mathematical 'object' is not an object in the sense that a physical object is, and has no more existence than the word 'object'---or any other abstract concept for that matter.

For the second, you wanted a citation of some kind. I can't imagine why a philosophical question needs a source of authority (literally logical fallacy) to answer, but if you want a source, I'll give you a mathematical philosopher: "I like mathematics because it is not human and has nothing particular to do with this planet or with the whole accidental universe - because like Spinoza's God, it won't love us in return." ~Bertrand Russell, 1912
We might not mean the same thing by 'mathematics', but anything I say now will be moot in a moment. And besides, there are whole books promoting that view with far more clarity than I can give at 2am.
(But if you want me to, I'll give an actual argument for this view when I'm more awake, which is probably what you really meant when you said citation.)

Scene break.

And now I give my present thoughts instead of just the thoughts you had wanted me to provide before. I'll be perfectly honest: not being anything like a physicist, the Tegmark essay on the Mathematical Universe Hypothesis seems very reasonable to my uninformed mind; and if true, it directly refutes several things I've stated above---and makes much philosophic language irrelevant, not that that's a new thing. But there is one problem particularly troubling to me: If this world is literally a particular mathematical structure, then how is it that we, being a part of that structure, can perceive other structures contrary to it? I get the feeling there's an answer to this that I simply haven't heard before. Footnote 17, for example, cites a paper addressing something similar.

In my defense, Tegmark specifically notes that things of the sort I've been stating are in line with the classical literature. And I still find the words 'empirical' and 'rational' to be useful in distinguishing things even if they could be, in principle, redundant.
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Re: Math discovered or invented?

Postby BlackSails » Sat May 29, 2010 6:55 am UTC

GyRo567 wrote:
Argency wrote:Sure, English is a construction, but when I talk about trees and rocks I'm talking about something real which I have discovered in the world and then come up with a word for. Same thing with triangles and the pythagorean.

The difference is that 'tree' and 'rock' denote things with objective existence. (In the sense of the material world) There is no such thing as a triangle in our universe. It is pure abstraction. Mathematics involves relations of ideas, not matters of fact about our particular world. If it were otherwise, we would have another empirical science, not a rational system. It would not be a purely deductive endeavor.



Tree and Rock denote ideas, not objects. "That tree" denotes an object with physical existance, but not all trees are that tree. We may speak perfectly well of a glowing orange tree, even though I doubt that any such tree exists.

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Re: Math discovered or invented?

Postby gmalivuk » Sat May 29, 2010 3:12 pm UTC

Definitions and axioms in mathematics may be invented, but then the consequences of those are discovered or revealed, because they exist logically and objectively as soon as the foundations are set up, even though we don't know about them yet.
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Re: Math discovered or invented?

Postby Yakk » Sat May 29, 2010 3:23 pm UTC

Imagine an incompressible universe.

A universe in which, for any portion of it, any "simpler abstraction" of it leaves a "remainder" that is about as complex as what you started with.

Our universe is extremely compressible -- you can talk about the properties of piles of apples have that piles of rocks have via counting that simplifies the situation massively. Natural numbers is a compression, in a sense, of the properties of things that can be counted. The concept of "apple" is another compression -- there are these things called "apples" that share properties with each other, and if you describe something as an "apple" then talk about how it differs from the "apple" ideal, you can describe the apple easier than if you tried to describe it "raw".

A universe in which this is not possible is a strange universe from our point of view. There are no nearly-repeating patterns at all. Things like the Pythagorean theorem cannot apply, because that means that describing two distinct routes between two different points contains redundant information.

In a universe that admits some kind of compression, you'll be able to generate an abstract system that approximates the behavior of the universe that is simpler than the universe -- it is a universe that admits maps that are smaller than the territory.

Mathematics builds maps that may or may not correspond to any territory. The maps they build are elegant and beautiful in some sense, and then science tries to find territory that is mapped by the output of mathematicians.
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Re: Math discovered or invented?

Postby supremum » Sat May 29, 2010 3:51 pm UTC

Argency wrote:
Cleverbeans wrote:
Spoiler:
Except that triangles don't exist in the real world, they're merely simplifications of nature we're capable of articulating. Mathematical objects are in general defined, their dependency on language should be obvious from that alone. I'd say that mathematical objects are more like unicorns, they bear enough of a resemblance to horse to be practical (I can ride a unicorn, therefore I can ride a horse, etc.) but it's too far reaching to say they exist.

When you say "I can ride a horse", you're not talking about any particular horse that exists. If you were, you'd have specified which one. What you're talking about is the ideal of a horse - the bit of a horse you'd have left if you took away all its individual quirks (like having a white spot on its nose or being called Pharlap, etc). I agree that that "ideal" of horsey-ness exists, and that "being rideable" is one aspect of it. Its also true that there exist tangible instantiations of horsey-ness - each and every horse in the world consitutes one. So its true to say "horses exist" (because you can touch them) but its also true to say "a horse can be ridden", even though you're only talking about the idea of a horse, not any specific, real horse. It's also true that unicorns are rideable (because that's part of being a unicorn), but it isn't true that unicorns exist.

Similarly, even if triangles weren't instantiated in the real world, it would still be true that [math]a^2 + b^2 = c^2[/math] because that's an inherent part of trianglehood, not something that relies on the specific properties of instantiated triangles. I don't agree that triangles aren't instantiated, though, because any three points in space that don't lie on the one line can be joined by three, forming a triangle. Triangles don't have to be made out of anything because there's no building material specified by the concept of trianglehood, so it doesn't matter if there's nothing actually occupying those points in space: its still true that the pythagorean holds for any three points which constitute a right-angled triangle. Even if humans weren't around to think about triangles all this would be true.


I don't find this argument very convincing because Euclidean Space is fundamentally a human invention. It may model reality fairly well at certain points, but Euclidean space can't really be said to physically "exist."

Pretty much by definition, mathematics can't be entirely discovered. It's pretty much impossible to do mathematics without postulates/axioms/undefined terms, and I don't think it's possible to come up with a convincing argument that there is some completely "natural" set of postulates/axioms/undefined terms that would lead to mathematics.

However, I also agree that it sounds a little silly to say that the pythagorean theorem was "invented," because once the axioms to describe Euclidean space are written down, it's possible to derive the pythagorean theorem from those axioms, so in some senses it counts as a discovery. So I guess I agree with gmalivuk.

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Re: Math discovered or invented?

Postby gmalivuk » Sat May 29, 2010 4:02 pm UTC

supremum wrote:I don't find this argument very convincing because Euclidean Space is fundamentally a human invention. It may model reality fairly well at certain points, but Euclidean space can't really be said to physically "exist."
Who said anything about physically anything?

I don't think it's possible to come up with a convincing argument that there is some completely "natural" set of postulates/axioms/undefined terms that would lead to mathematics.
But mathematics isn't just what logically follows from one particular set of axioms. (I disagree with your claim that there need to be undefined terms.) It's the structure of what follows from any specified set of axioms. And that structure doesn't depend on us first specifying which part of it we're looking at.

It's somewhat like saying that some treasure really objectively exists, and anyone can equally verify that it exists once they've been given the same map.
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Re: Math discovered or invented?

Postby BlackSails » Sat May 29, 2010 4:22 pm UTC

supremum wrote:I don't find this argument very convincing because Euclidean Space is fundamentally a human invention. It may model reality fairly well at certain points, but Euclidean space can't really be said to physically "exist."


Sure it can.

At some point here, space has positive curvature. At some other point there, space has negative curvature. By rolle's theorem, there exists at least 1 point in between with zero curvature.


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