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[P]
Is mathematics invention or discovery?

By StrontiumDog in Culture
Fri Jan 05, 2001 at 01:15:44 PM EST
Tags: Science (all tags)
Science

Mathematics has often been humorously defined as "what mathematicians do". But what do mathematicians actually do?

Are they explorers, venturing out into virgin terrain and making hitherto unknown discoveries? Do all mathematical discoveries lie "out there", placed courtesy of an unknown deity for the edification of human mathematicians?

Or are they engineers, constructing theorems on a scaffolding of axioms and logical propositions? Is mathematics utilitarian, where abstract patterns are mapped onto existing physical structures, with only the patterns that make any physical sense being termed valid?


The Greek philosopher Plato postulated that geometrical entities had an existence separate from this world: the triangles, circles, squares we encounter in our daily lives are imperfect images of the perfect mathematical objects they represent.

This view is further expounded on by mathematician Roger Penrose. In The Emperor's New Mind he states

When mathematicians come upon their results are they just producing elaborate mental constructions that have no actual reality, but whose power and elegance is sufficient to simply fool even their inventors into believing that these mere mental constructions are "real"? Or are mathematicians really uncovering truths which are, in fact, already "there" - truths whose existence is quite independent of the mathematicians' activities? I think that, by now, it must be quite clear to the reader that I am an adherent to the second, rather than the first, view, at least with regard to such structures as complex numbers and the Mandelbrot set.

But is this so? Are mathematical discoveries not more akin to invention? Take chess, for instance. Given a few arbitrary rules an almost infinite number of chess games can be constructed. Each chess game is not arbitrary: not all combinations of pieces on a chess board are valid. All valid games can however be derived from a small number of arbitrary man-made rules.

When one looks at axiomatic set theory, for instance, the similarity with chess is striking. Starting from a few axioms (fundamental unprovable assertions) a whole range of mathematical results can be built up. For instance, Peano's axioms generate the natural numbers. The French mathematical group Bourbaki is attempting to take this to its logical conclusion by unifying all mathematics using axiomatic set theory: an undertaking very similar to the herculean Principia Mathematica project by Russell and Whitehead in the early 1900's.

In addition, if mathematical truth is Platonic, where then does it exist? Quantum physicists insist that at the most fundamental level Nature is statistical, and absolute, deterministic laws do not exist. Yet there is nothing more fundamental, or deterministic, than mathematics itself. If mathematics be Discovery, then what aspect of our non-deterministic universe are mathematicians discovering?

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Poll
Mathematicians are
o Discoverers 25%
o Inventors 5%
o Both 36%
o Pencil-necked geeks 32%

Votes: 74
Results | Other Polls

Related Links
o Plato
o The Emperor's New Mind
o Peano's axioms
o Bourbaki
o Principia Mathematica
o Also by StrontiumDog


Display: Sort:
Is mathematics invention or discovery? | 52 comments (48 topical, 4 editorial, 0 hidden)
Can be both ways (3.62 / 8) (#3)
by tftp on Fri Jan 05, 2001 at 10:52:08 AM EST

The question appears to be invalid because it assumes that the object of research (math) can be in only one of few well-defined states (invention -or- discovery). However it is obvious that sometimes math is a science (where discoveries are made), sometimes it is a technology (where new products and methods are invented).

A compression algorithm can be invented (by applying process of invention - have a goal, research existing solutions and propose an improvement). A formula can be discovered by applying scientific process (a natural phenomenon, want to know, have a theory, run experiments, prove the theory).

Only one way (2.33 / 3) (#13)
by StrontiumDog on Fri Jan 05, 2001 at 12:02:52 PM EST

The question appears to be invalid because it assumes that the object of research (math) can be in only one of few well-defined states (invention -or- discovery). However it is obvious that sometimes math is a science (where discoveries are made), sometimes it is a technology (where new products and methods are invented).

At the operational level, sure. At the fundamental level it's a dichotomy: it's either one or the other, and you can't have it both ways. Maybe the terms "discovery" and "invention" are misleading. Let me put it another way. Take a compression algorithm -- say run-length encoding. The expression of RLE in source code, such as in bitmap compression, is clearly invention. This is at the operational level. But the concept of RLE is an abstract one: this is the fundamental level. Many programmers have independently thought up their own RLE schemes over the past decades. The form in which these schemes are expressed differ: byte-run encoding, integer encoding, special markers indicating the start of runs etc. The principle behind all these RLE schemes is the same: compression by replacing a series of identical values with the value itself and the number of repetitions.

This becomes particularly clear when mathematical, programming or logical concepts can be subjected to taxonomy: someone looks at a piece of code and says "Hey, this looks like some kind of run-length encoding scheme". The recognition of an underlying abstract pattern, distinct from the expression of that pattern, is what I'm getting at. And the simple dichotomy is, did this pattern exist a priori (or before the code in question was written) or a posteriori (only after the code was written, and only in the mind of the beholder).

[ Parent ]

Recount, please! (2.66 / 3) (#19)
by tftp on Fri Jan 05, 2001 at 12:37:47 PM EST

At the fundamental level it's a dichotomy: it's either one or the other, and you can't have it both ways.

Indeed, every single case examined must fall into one of those categories. However the article asks about the science of mathematics as a whole. If all test cases belong to mathematics and if some cases are discoveries and other are inventions then we can not say that math as whole is a discovery or an invention. Reasoning behind our classification does not matter.

[ Parent ]

Dichotomy, schmichotomy (3.00 / 4) (#24)
by 0xdeadbeef on Fri Jan 05, 2001 at 01:10:54 PM EST

And the simple dichotomy is, did this pattern exist a priori (or before the code in question was written) or a posteriori (only after the code was written, and only in the mind of the beholder).
Which is itself impossible to objectively determine. This degenerates into an almost Zen-like argument: if no one is there to understand the pattern, does the pattern exist? Did the lever exist before humans invented it, or does the use of the principle in nature make it exist? As far as I know, nature hasn't made use of run-length encoding. But if has, does that use discount the achievement of the first human who recognized it?

[ Parent ]
Discovery (3.90 / 11) (#5)
by DaFlick on Fri Jan 05, 2001 at 10:56:16 AM EST

I'm a Maths undergraduate (note: Maths with an s - I'm British!), and so far, from what I've learnt, the progression of mathematics has been a series of discoveries, not inventions.

Many physicists, and indeed a lot of notable philosophers, believe that the universe has an underlying strict mathematical basis (string theory etc). Therefore mathematics is all about discovering the truth - what is real. Now, I freely admit that we have to begin by clutching at a few straws - we state a few axioms - the axioms of real mathematics for example - which we then go on to use to prove other theorems.

Now, my main argument that mathematics is discovery, not invention, is that the mathematical theorems and formulae exist before we discover them - Pythagoras didn't come up with his famous theorem and then it was true, the square on the hypontenuse of a right angled triange was equal to the sum of the squares on the other two sides before he happened to find this out. Therefore the theorem was not invented - he didn't create it, he merely discovered it.

Quantum physicists may insist all they like that the basis of nature is statistical. However, the interactions of probabilities tend towards complete predictability as the number of probability events approaches infinity. Take for example the central limit theorem - predictability can be built out of randomness, and as there are a huge amount of particles involved in quantum physics, we can predict overall outcomes, at a much higher level such as the movement of planetary bodies because of this funneling effect.

A simplified example: An unwieghted, six sided die has a probability of 1/6 of producing each number. We have a rectangular distribution. From this, you have as much chance of getting a 6 as a 1 or a 3. Now introduce a second identical die. Add the two numbers obtained together. Now we have a greater chance of a 7 (1/6) than a 12 (1/36). It's becoming slightly more predictable. Adding more and more dice to the system does broaden the range of results, but the resulting distribution tends to being a normal distribution as the number of dice tends to infinity. The relevance? We can predict the number of results of each value as the number of rolls of the dice tends to infinity, and as the number of rolls of the dice does tend to infinity, the probability of a normal distribution perfectly formed tends to 1. So as the number of random events becomes infinite, the outcome changes from being a probability to a certainty.

The argument could also be taken as discovery of a universe which is governed by these laws - we state that "If these axioms are correct, then ..." and discover the nature of a completely seperate universe to our own which follows these laws.

A famous quote to finish with: "Mathematics is the language of nature". Now let nature be defined by mathematics for this argument: Nature is defined by mathematics, mathematics is the language of nature. Nature was created by God (if you believe) and hence God used the language of mathematics. So by studying the effects of mathematics we can understand nature, but by studying mathematics itself, perhaps we can begin to glimpse God...

OK, not exactly a perfect argument for the study of maths, but the best I could come up with on the spot ;-)

The reason this is "unclear" (3.00 / 8) (#6)
by trhurler on Fri Jan 05, 2001 at 10:58:42 AM EST

The reason this seems unclear is quite simple: mathematicians are good at mathematics, but this does not make them good epistemologists. Mathematics, fundamentally, has only one real rule - logic. Given that fundamental rule and some assumptions(which in fact, while not provable, are probably verifiable via other means for all intents and purposes,) you can construct all of mathematics - given infinite time and patience. The mathematics is implied by the logic(which is not to say that any given part of it is obvious given only the premises needed to derive it,) - so the question to ask is really whether or not logic was invented or discovered. The probable truth is that it is an inherent part of being people, and was neither truly discovered nor invented, but merely identified formally despite having been in use all along. This is not to say it does not reflect a pattern(non-contradiction) which is found in the real world - but it IS to say that people do not really have a choice in the matter - they are using logic long before they've ever heard the term, even if only implicitly.

--
'God dammit, your posts make me hard.' --LilDebbie

There is a Godelian argument which refutes this (2.66 / 3) (#7)
by streetlawyer on Fri Jan 05, 2001 at 11:15:49 AM EST

Given that fundamental rule and some assumptions(which in fact, while not provable, are probably verifiable via other means for all intents and purposes,) you can construct all of mathematics - given infinite time and patience.

I'm not sure what you mean by "all of mathematics", but one of the things we know for certain is that there is no set of assumptions (axioms) from which you can deduce all the theorems of even such a subset as elementary number theory, even if you do have infinite time and patience.

--
Just because things have been nonergodic so far, doesn't mean that they'll be nonergodic forever
[ Parent ]

strange loops (3.00 / 2) (#10)
by CodeWright on Fri Jan 05, 2001 at 11:33:21 AM EST

very true.

godel also demonstrated, in the same vein of thought (ie, self-referential systems), that logic was an insufficient framework with which to define logic itself.

the most widely recommended layman's introduction to these ideas is probably Douglas Hofstadter's Godel, Escher, Bach : An Eternal Golden Braid



--
A: Because it destroys the flow of conversation.
Q: Why is top posting dumb? --clover_kicker

[ Parent ]
Confused (4.00 / 1) (#22)
by Simon Kinahan on Fri Jan 05, 2001 at 12:52:54 PM EST

I'm not sure what you mean by "all of mathematics", but one of the things we know for certain is that there is no set of assumptions (axioms) from which you can deduce all the theorems of even such a subset as elementary number theory, even if you do have infinite time and patience.

I think the above is confused. Godel proved that some (intuitively true) statements are unprovable in a consistent, finite formal system. You can however deduce all theorems of number theory from a finite set of axioms, as that is what theorems are by nature - Godel only proved number theory was necessarily incomplete (not able to prove or disprove all statements)

Simon

If you disagree, post, don't moderate
[ Parent ]

aargh (none / 0) (#26)
by streetlawyer on Fri Jan 05, 2001 at 01:13:05 PM EST

sorry -- for "theorems" read "true sentences".

--
Just because things have been nonergodic so far, doesn't mean that they'll be nonergodic forever
[ Parent ]
Godel anyone? (4.00 / 2) (#9)
by spiralx on Fri Jan 05, 2001 at 11:27:25 AM EST

Given that fundamental rule and some assumptions(which in fact, while not provable, are probably verifiable via other means for all intents and purposes,) you can construct all of mathematics - given infinite time and patience.

Errm, maybe I've completely missed what you're trying to say, but Godel demonstrated that within any given mathematical system defined by a consistent set of axioms there will always be prepositions that you cannot decide whether they are true or false using the rules and axioms of that system alone. Hence even with an infinite amount of time and patience, you cannot ever complete a mathematical system in its entirety.

You're doomed, I'm doomed, we're all doomed for ice cream. - Bob Aboey
[ Parent ]

Godel (2.00 / 2) (#12)
by trhurler on Fri Jan 05, 2001 at 11:51:04 AM EST

What Godel proved was a lot more limited than most readers seem to think. He did not prove that "some mathematical statements are unprovable." He did prove that that is true given a fixed set of axioms. This is why I said "some" axioms. Of course, a good part of the undecidable statements are also uninteresting, and furthermore, a good part of the remainder are either reasonable axioms in and of themselves or suggest the possibility of further axioms.

That said, I stand by the point of what I said, which is that mathematics is the application of something that existed before we recognized it as such.

--
'God dammit, your posts make me hard.' --LilDebbie

[ Parent ]
But (none / 0) (#14)
by spiralx on Fri Jan 05, 2001 at 12:09:51 PM EST

What Godel proved was a lot more limited than most readers seem to think. He did not prove that "some mathematical statements are unprovable." He did prove that that is true given a fixed set of axioms.

But if you were trying to establish the internal consistency of a mathematical system such as elementary arithmatic, then that is exactly what Godel showed you cannot do without introducing new axioms and thus failing to prove the consistency of what you started with.

Of course, a good part of the undecidable statements are also uninteresting, and furthermore, a good part of the remainder are either reasonable axioms in and of themselves or suggest the possibility of further axioms.

Yes, but many important questions are, such as whether C, the number of real numbers, was the same as aleph-one (aleph-0 to the power of aleph-0). In the end, it was shown that this statement was undecidable and became a new axiom instead. Or the parallel-postulate axiom of Euclidean geometry.

That said, I stand by the point of what I said, which is that mathematics is the application of something that existed before we recognized it as such.

I don't disagree with that, merely your statement about infinitity and provability...

You're doomed, I'm doomed, we're all doomed for ice cream. - Bob Aboey
[ Parent ]

Not quite right (4.00 / 1) (#21)
by Simon Kinahan on Fri Jan 05, 2001 at 12:49:00 PM EST

Godel proved that if you have a formal system with a fixed set of axioms that are sufficient to encode ordinary arithmetic, there will either be an inconsistency, or there will be statements that are undecidable within the system. You can prove that the formal system is internally consistent (cannot prove any pair of contradictory statements) provided you accept that some statements are undecidable.

Simon

If you disagree, post, don't moderate
[ Parent ]
True, true (none / 0) (#18)
by 0xdeadbeef on Fri Jan 05, 2001 at 12:32:42 PM EST

If we ever develop a Grand Unified Theory from which all physical laws can be derived, one can claim that all possible inventions exist, and therefore we don't create them, we simply find them.

I believe that, but it doesn't mean I don't recognize the effort and skill involved in finding those inventions. This division between "eternal truth" and "human invention" does not exist, either in the real world or in the pure abstractions of mathematics.

[ Parent ]
It seems to me.... (3.50 / 6) (#11)
by 11223 on Fri Jan 05, 2001 at 11:34:24 AM EST

Consider a universe with finite amounts of matter and energy, such as ours. In this universe, there is a maximum representable number, because at some point you would run out of matter or energy with which to represent this number. Thus, you could probably safely say that there is a maximum integer in our universe.

The question you seem to be asking seems to be whether the irrationals are rule-based (rational!) or not. If they're not based on any rules, then it's all discovery. If they are based on rules, it's invention. Well, we can prove that the irrationals aren't rule based - if they were, they'd be countable, and I'm sure you're aware of Russel's proof. (I hate that proof.) I'll try to satisfy you, however, by making them rule-based in another universe.

Call the maximum integer in our universe Nu (read N sub U. I hate plain ascii for math!) We can define a field, such as Z mod (Nu+k), (Nu+k is prime) under which the exponent of two large but representable numbers leads to a third, unpredictable number (because Nu+k is not representable, and thus we can do no computation involving it.) It's my pet theory that the irrational numbers are somehow resultant from this - irrational to us, but the ideas from which they come (such as the operation of a Z mod P field) are perfectly understandable to us.

Consider then that when we start turning large chunks of the universe into a simulator for another universe, the rules upon which we make it will probably be constructed in a similar fashion - we'll use N for that universe to make irrational numbers that are perfectly predictable for us.

Someday, in this simulated unverse, another mathemetician will come up with a pet theory about how the irrationals in his unverse are constructed, and the cycle will be complete....

--
The dead hand of Asimov's mass psychology wins every time.

interesting (none / 0) (#15)
by rebelcool on Fri Jan 05, 2001 at 12:17:50 PM EST

perhaps thats how our little universe came to be...

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[ Parent ]

The 13th Floor (none / 0) (#35)
by jabber on Fri Jan 05, 2001 at 04:17:02 PM EST

Not an entirely bad movie about how Joe Hollywood sees this sort of discussion. Rent it to kill a few hours.

[TINK5C] |"Is K5 my kapusta intellectual teddy bear?"| "Yes"
[ Parent ]

seen it (none / 0) (#37)
by rebelcool on Fri Jan 05, 2001 at 05:38:56 PM EST

it was alright.

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[ Parent ]

Representable? (none / 0) (#17)
by StrontiumDog on Fri Jan 05, 2001 at 12:30:56 PM EST

. In this universe, there is a maximum representable number, because at some point you would run out of matter or energy with which to represent this number. Thus, you could probably safely say that there is a maximum integer in our universe.

Why should there be a maximum representable integer? I have a few problems with such a statement:
- it assumes the universe is finite
- what exactly do you mean by "represent"?
- what prevents me from calling any number I wish (for instance the maximum integer) "googleplex" and doing computation with numbers offset from "googleplex"? Or to use a less contentious example: transcendental numbers. The decimal representation of pi is infinite, but that does not prevent pi from being a well-defined, eminently usable number.


[ Parent ]

Well.... (none / 0) (#20)
by 11223 on Fri Jan 05, 2001 at 12:38:32 PM EST

Well, represent might have been a bad word. However, you can't do all your computation by offsets from googlplex; as soon as you need to do addition or subtraction with your multiplicitave identity your whole little scheme is shot as you now need to represent your googleplex as a sum of ones.... which you can't do.

Secondly, you can't get any useful results except for identities from pi. All useful results come out of an approximation of pi by a rational number.

--
The dead hand of Asimov's mass psychology wins every time.
[ Parent ]

The problem (4.00 / 1) (#27)
by StrontiumDog on Fri Jan 05, 2001 at 01:13:09 PM EST

. However, you can't do all your computation by offsets from googlplex; as soon as you need to do addition or subtraction with your multiplicitave identity your whole little scheme is shot as you now need to represent your googleplex as a sum of ones.... which you can't do.

Regardless of the representation, a lot of mathematics can be done. Or so we think. To quote Greg Chaitin, in "The Limits Of Mathematics" (Springer 1997, pp 85):

You're young, you're trying to learn mathematics, you're doing elementary number theory, and you don't really start to worry about the fact that elementary number theory presupposes arbitrarily large positive numbers, and how do they fit in the universe? Imagine a positive integer that's 10^(10^10^10))) digits long. Does it exist? In what sense does it exist? You don't care -- right? -- you prove theorems about it, you know that it would be commutative, right? a + b is going to be equal to b + a even if neither number fits in the universe! [Laughter] But then later on in life you start to worry about it! [Laughter]
It's a very interesting book, I recommend it highly, and no Chaitin doesn't give a definitive answer although he leans towards the theory of mathematics as human artefact. But in classical mathematics, representation is generally irrelevant. Computation with the "googleplex" number I mentioned is eminently possible in mathematics: messy perhaps, but entirely possible.

[ Parent ]
Quantization of spacetime (none / 0) (#25)
by spiralx on Fri Jan 05, 2001 at 01:12:11 PM EST

It does seem as though string theory is going to leave us with a spacetime that is one sense quantized. This is tied to duality theory, which relates two or more seemingly different quantum theories, in this case string theories. Whilst these theories may have totally different classical limits (when Planck's constant, h-bar, is taken to zero) they are in a sense the same quantum theory.

What does this mean? Well say in one theory you have the radius of a compactified dimension being related to 1/K, where K is a coupling constant. Since K can decrease without limit then this isn't quantized, right? But due to a dual relationship with another string theory, this radius can also be expressed as being proportional to K! And what's more, the cutoff point where the K and 1/K relationships meet is the Planck length.

It's all pretty complicated, but the point is that whenever you encounter a coupling constant leading to a length shorter than the Planck length, you can use duality to convert is to an equivalent theory where the coupling constant provides a different length greater than the Planck constant. Both are equally correct, and hence in that sense, space can be said to be quantized.

There is a similar argument for time I believe, with the Planck time being 10^-43s and the Planck length being 10^-33cm. If you take these as the smallest meaningful units of measurement, then yes, there is only a finite amount of information that can be stored in the Universe, and hence a finite maximum representable integer.

You're doomed, I'm doomed, we're all doomed for ice cream. - Bob Aboey
[ Parent ]

Representable stuff (none / 0) (#33)
by tftp on Fri Jan 05, 2001 at 03:08:46 PM EST

As I understood, "representable integer" here has everything to do with the size of a set of all integers you care about. Offsets etc. do not matter, just make sure that you can tell the difference between any integer and any other integer. I would assume that you need to store some information about those integers; in best case you need as little as one electron (or something like that), as long as you can read the state of that electron (which may be a problem on its own). Then the total number of electrons in Universe tells you how many bits of storage you have. If you exceed that number you can not compare two integers simply because you don't have enough storage to hold the unique IDs of those integers in your set!

[ Parent ]
Discovery.. (3.00 / 5) (#16)
by rebelcool on Fri Jan 05, 2001 at 12:23:24 PM EST

An invention is something wholly new that didnt exist before, that one as built.

However, in math nothing physically exists, therefore all numbers and equations and their infinite combinations already do exist in the mathematical abstraction of the universe. Thus when a new formula or rule appears, it has been discovered because it was there the whole time, but just not noticed.

Think of it as finding a new insect in the rainforest that noone had ever seen before. You did not invent this insect, of course. You discovered it. It was there the whole time, just never noticed.

I'm no mathematician (heck, i'm really bad at calculus..) but thats just the way I see it.

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A question for young grasshopper (none / 0) (#28)
by 0xdeadbeef on Fri Jan 05, 2001 at 01:51:30 PM EST

Did the Bessemer process for refining iron into steel exist 10^-30 seconds after the Big Bang?

[ Parent ]
well (none / 0) (#30)
by rebelcool on Fri Jan 05, 2001 at 02:27:00 PM EST

i know nothing of refining steel though I have heard of that...

The mathematical part of it would be yes... when man first added 2+2 together, he did not "invent" this expression. The number 2 already existed mathematically, as did the concept of addition.

Since math is not "real", but rather an abstraction, things like infinity can exist. Therefore, infinite combinations of numbers and the various ways to act on them already exist in the mathematical sense.

I can't really think of any good physical-world ways to compare it since infinity doesnt truly exist (that we know of anyways).

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[ Parent ]

There is the infinite in every plum blossom (none / 0) (#44)
by 0xdeadbeef on Sat Jan 06, 2001 at 01:09:35 PM EST

But the universe, while finite, is ruled by the same mathematics. In a similar way, you could say the very existence of the universe implies the existence of all possible permutations from its initial state, as they are all attainable by following the ramifications of the basic laws.

So in a sense, the process of invention, even of physical machines, is a process of sifting through potential "truths" to find those that are useful, which is exactly what many mathematicians do.

Of course, we don't really know if there are basic laws from the universe can be extrapolated. So this could be nonsense. :-)

Software has been written that finds useful mathematical truths. Software has been written that, while very primitive, finds useful machines. And natural selection has been doing both since the beginning of time. Do these things invent anything, or are they merely discovering that which is already there? How is what we do any different? Is there anything special about "intelligence" doing the inventing, and can that difference be proven or demonstrated?

[ Parent ]

true that (none / 0) (#45)
by rebelcool on Sat Jan 06, 2001 at 01:18:09 PM EST

It's quite possible that everything has been created already and we just don't know about it. In that case, I would think we're still more likely to "discover" things as opposed to inventing them.

Perhaps we should divide it into nature vs. man-made. Although everything man-made is made from natural things, we'll ignore that paradox for the moment.

Math is of course, natural. It's been here since the creation of the universe (or before..who knows), and will be here until the end of the universe. Completely indepedent of man. The same is true for the ways the atoms interact with one another, laws of physics and so on. Since these truths have been here all along, they are "discovered".

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[ Parent ]

What do Mathematicans do? (3.00 / 7) (#23)
by guffin on Fri Jan 05, 2001 at 12:56:10 PM EST

To paraphrase Paul Erdos: 'Mathematicians are machines that turn coffee into theorems'

Invention.. (3.71 / 7) (#29)
by balls001 on Fri Jan 05, 2001 at 02:20:43 PM EST

Do all mathematical discoveries lie "out there", placed courtesy of an unknown deity for the edification of human mathematicians?
If possibilities that lie "out there" are automatically discoveries, then the lightbulb or car were never invented, but rather the way to assemble one was at some point discovered by a human.

The answer is "no". (3.75 / 8) (#31)
by error 404 on Fri Jan 05, 2001 at 02:54:28 PM EST

It isn't really either. Math is math. A mathematical effort can be made in an inventing frame of mind or in an exploratory frame of mind, or better yet, in a state of mind specific to math.

Kind of like the "is an electron a wave or a particle" question. DUH - an electron is an electron, an item that has no exact analog in the directly experienced world. An electron acts wavy in some ways and particly in some ways. But it isn't a wave and it isn't a particle.
..................................
Electrical banana is bound to be the very next phase
- Donovan

Platonism in Chess (3.60 / 5) (#32)
by kallisti on Fri Jan 05, 2001 at 02:57:20 PM EST

Your example of chess is actually a good argument in favor of Platonism.

There are a large number of possible positions in chess, and there are a large number of impossible positions. Which positions are possible is determined by the rules, which is an arbitrary set of rules. The way these rules "unfold" into the game positions is not arbitrary, however. There are structures, such as relative strengths of positions, that aren't explicit in the rules, but can only be determined by playing out many, many games.

Most of math isn't the setting of the rules (axioms), it is in the unfolding. Admittedly, the rules are chosen for maximum value in unfolding, but some amazing structures form even with few axioms. For example, the idea of groups, about as fundamental an idea as can be found in math, led to the "Monster" an immense group that represents a set of rotations in a space of about 160000 dimensions, and is the largest finite group not divisible into smaller groups. To me, the "Monster" existed as soon as Galois came up with groups, it just took years to find.

Math has all kinds of oddball things that are implicit in the original axioms, but need centuries of work to extract. I guess one difference between Platonists and non-Platonists (Constructivists or Formalists) is dependent on what you consider math to be. If you consider math to be finding the structures inherent in a set of axioms, then I would say that was Platonist. If you think math is finding axioms that concisely explain various structures and things found in the Universe, then you are a Constructivist. If you think it is all a fun game, and the fact of its extreme usefulness is basically a coincidence, that sounds Formalist.

A modern Principia Mathematica on the web (3.66 / 6) (#34)
by joshpurinton on Fri Jan 05, 2001 at 03:47:58 PM EST

The Metamath Proof Explorer has 60 MBytes of web pages containing over 3000 completely worked out proofs in logic and set theory, interconnected with more than 130000 hyperlinked cross-references. Each proof is pieced together with razor-sharp precision using simple rules, allowing almost anyone with a technical bent to follow it without difficulty. With point-and-click links, every step can be drilled down deeper and deeper into the labyrinth until axioms will ultimately found at the bottom. Armchair mathematicians can spend literally days exploring the complex tangle of logic leading, say, from 2 + 2 = 4 back to the axioms of set theory. The proof collection includes many famous theorems of elementary set theory.

Math is a way of looking at the Universe (3.00 / 6) (#36)
by Tau Neutrino on Fri Jan 05, 2001 at 04:49:52 PM EST

Math is not the language of the Universe. Whatever that language is, it's beyond mere human understanding. That's not to say that math isn't very good at describing some of the objects and processes of the Universe. It is; one of our best.

But art is another way of looking at the Universe and describing it to other humans. It's also very useful, although not so highly valued in Western society today. And religion is yet another way.

Each of these (and many more) is a human invention: created and used by millions of humans to help make sense of their experience in the Universe, and to share it with other humans. That's why they will always keep changing, growing, and shifting; as long as we do.

--
Theater is life, cinema is art, television is furniture.
Music and Math (4.00 / 7) (#38)
by bukowski on Fri Jan 05, 2001 at 06:25:46 PM EST

Just an aside on the create/discover theme with regards to music. Everyone I'm sure is familiar with mp3's. Being a computer file, these are stored as a string of 0's and 1's - a binary representation of a number. This number has the property that, when fed through an mp3 decoder, it produces a song. Well, that number was there before anyone even wrote the song!So even if Frank Zappa had never wrote "Peaches en Regalia", if the number that corresponds to the song were fed through an mp3 decoder, there it would be. Not only did every song that Frank Zappa wrote already exist before he wrote it ( as a number ), but every song that he didn't write is there too! Because if he had wrote (sung) it, it would be able to be converted into a number.

Of course, these numbers are to big to iterate through, but imagine if you could. You could "discover" Zappa doing a cover of The Hanson's "Mmm-bob", because it would correspond to a number.

Same goes for works of art, literature, etc.

Invention by Enumeration... (4.00 / 1) (#48)
by Morn on Sun Jan 07, 2001 at 06:12:40 PM EST

...or should that be 'Discovery by Enumeration'?

every song that he didn't write is there too

Apart from those containing sounds unrepresentable by MP3.

My main point in replying is not to be pedantic, however, but is to MLP to here - this guy has made an electronig objerct which you can hang on your wall - it enumerates all possible monochrome 8X8 images. You can also download a program for BeOS to sit on your desktop and do the same

Possibilities like these have always intrigued me - I'd like to write a 'Valid ELF Binary' enumerator and hope it comes up with the next killer app - I just don't fancy having to sit testing all the billions of billions of output programs to find it :-)

[ Parent ]

quantum (2.75 / 4) (#39)
by andyschm on Fri Jan 05, 2001 at 08:34:45 PM EST

Really the question comes down to physics and the quantum-view of the universe.

If you believe that the universe is determinstic, then "invention" is strictly speaking, impossible. I can't invent a theorem any more than someone can "invert" a spoon or a car. As the universe progresses it is only a matter of time until the molecules arrange themselves into a spoon or a car or the electron-firing pattern produces a mathematical theorem. So the question really has -nothing- to do with math, per say -- although math is a great topic and I love to talk about it (and study it... a lot!).

If you follow the quantum model, then everything is fuzzy and probabilstic, and I think this would be especially true for the human mind, which is basically a big fuzzy nano-computer. In this case I suppose we would call "new" things "inventions".

Whatever. "Geek" works too.


I think it depends (3.50 / 4) (#40)
by (void *)0x00000000UL on Fri Jan 05, 2001 at 09:00:56 PM EST

I'm speaking as an engineering student. I think mathematics are a pure invention of the spirit. Mathematics tries to map the universe into geometrics figure or equations. I don't agree with Platon: the triangles, circles, squares we encounter in our daily lives are perfect things modeled by imperfect simplified mathematical objects or equations. Mathematics is a model of the universe broken down into managable parts. This way we can predict to an acceptable precision how thing will go. Nature is infinitely complex and mathematics are here to simplify things. That's my view. It's normal that nature seems to act mathemically: we created mathematics to mimic nature's behavior. If mathematics were a discovery, we would not need basic unprovable axioms (well I think).

Some comments (3.33 / 3) (#41)
by SIGFPE on Sat Jan 06, 2001 at 12:03:02 AM EST

Quantum physicists insist that at the most fundamental level Nature is statistical, and absolute, deterministic laws do not exist
It certainly doesn't insist. Many physicists (more than half of well known physicists according to a survey in Scientific American a couple of years ago) subscribe to the relative state formulation of quantum mechanics which is 100% deterministic (depending on your definition of deterministic). It is not generally taught to undergraduates so it is not terribly well known outside of physics circles except in popular accounts where it goes by the misnomer "many worlds".

As for whether mathematics is invented or discovered. Think of a hammer. It's an invention. I whack a piece of metal and find it makes a pleasing sound. I have made a discovery and have invented a musical instrument. Where's the conflict between 'invent' and 'discover'? Mathematics is a tool like any other. It's a way of using pen and paper, or computers, or our fingers or even our neurons to do things like predict whether a bomb will blow up enough people to end a war or figure out where to render a polygon in your favourite video game. It's a particularly cool invention because there's lots about it to discover. Can you make clearer the distinction between 'invention' and 'discovery' because clearly you do see a conflict between these concepts but I don't.

As for Penrose's opinions. They're just opinions. We all have them. But he'd get marked down if he posted to Kuros5hin because he clearly has no argument to back them up.
SIGFPE

Determinism (4.00 / 1) (#43)
by Mad Hughagi on Sat Jan 06, 2001 at 03:16:24 AM EST

Good call on bringing up the "many worlds" concept. I guess the "hidden variable" theories might lay an assault on that comment as well.

The distinction that I would make between an invention and a discovery is that an invention is something that is "dreamed" or "thought up" by a person, while a discovery is something that is found elsewhere and recognized by man. I'm afraid it quickly tumbles into philosophy though, where do you make the distinction between the mind of a person and the rest of a reality?

I guess you can only have an invention (the Eureka, I've found it! type) if you believe that the human entity possesses something greater than that which can be found elsewhere in the universe (divine inspiration or some kind of mystical ability).

As far as I'm concerned it's all self-discovery, on every scale. To say that something is an invention implies that one (or more) of our mental constructs isn't apparently observed in the universe. This neglects the fact that our ideas themselves are patterns that came to be by the flow of the very universe itself - I guess making these distinctions is part of the fragmentation approach, but as you have put it, I don't really see the need to further classify mathematics.


HUGHAGI INDUSTRIES

We don't make the products you like, we make you like the products we make.
[ Parent ]

Determinism without many worlds... (none / 0) (#46)
by jason on Sun Jan 07, 2001 at 12:45:49 AM EST

There are deterministic possibilities without the many worlds hypothesis, as well. Another possibility is that there is a complicated deterministic process down there somewhere, but we either don't need to or cannot follow the turtles all the way down. It's a Jaynesian view, using probability and statistics to compensate for our lack of knowledge.

Jason

[ Parent ]

Difference between discovery and invention (none / 0) (#47)
by chui on Sun Jan 07, 2001 at 01:17:10 AM EST

Sure, there is a difference. For example, people discover islands, they don't invent them. I discovered some food in the refrigerator that I didn't know I had before. I think the clapper is a clear example of an invention. It was made with a very specific goal in mind, and all the technology to make it was already there. However, I do believe most of the time, they go hand in hand, and I believe math is one of those areas. Some things are discovery, other things are invention.

[ Parent ]
Discovery and Invention (4.25 / 8) (#42)
by Khedak on Sat Jan 06, 2001 at 12:38:39 AM EST

In addition, if mathematical truth is Platonic, where then does it exist?

Platonic views are simplified, but in essense discovery and invention aren't that far removed. An invention is simply a discovery of something that works for a particular purpose. For example, are airplanes a discovery or an invention? Well, the principles on which they operate had to be discovered before they could be invented. And sometimes vice versa, working inventions lead to discoveries. They go hand in hand, and you seem to ask whether mathematics is an "invention" as if this is in opposition to a discovery.

Using chess as a parallel to mathematics is fair, because chess is highly mathematical. The fact is, many mathematical constructions that may look useless at the outset in fact later are shown to have parallels in the real world. You're right in that it's an important question (perhaps the most important question) of whether mathematics and the universe are inherently related, or if such relations are coincidence. Geometry and calculus have innumerable real-world applications, whereas more abstract concepts like the proof of Fermat's last theorem seem not to. Then again, there's the question of what constitutes application to the "real world." Again, Plato has his own ideas about what is real and what is not, but that's a very deep philosophical question that is closely tied to this one.

It's important to remember that mathematics has particular rules that have been selected for a reason, usually because they are logically drawn from other logical axioms. For example, (as Hofstadter points out in his book Goedel, Escher, Bach which many k5 readers seem to be familiar), when the concept of hyperbolic or parametric (as opposed to Euclidean) geometry was proposed, it was denounced because it didn't "fit the real world." However, there are applications for which these geometries are very useful. Hoftstadter seems to think that these geometries are irreconciliably different, and that the fact that more than one can be made to fit the world means that there is no "true" geometry. But many believe that in fact the aspects that these geometries share make up what is "real", since they all differ in only their most abstract axiom.

You seem to forget all the uses of mathematics when you ask what aspect of our universe mathematicians are discovering. Citing quantum mechanics as evidence of the non-reality of mathematics is extremely ironic: quantum mechanics is one of the most heavily mathematical physical sciences there is. In that light, your question needs a little more clarification. What exactly do you want of mathematicians? Do you want them to say "Although all other physical sciences, which have real world applications, rely in some form on mathematics, mathematics itself has no intrinsic truth value." While you could make such an argument, what would be the point?

In essence, you're asking people to prove that mathematics exists in the same fashion that atheists ask the faithful to prove that God exists, or nihilists ask people to prove that anything at all exists. It's easy to deny the existence of anything at all, since any time someone presents you with evidence for its existence, you can question the reality of that piece of evidence, and so forth ad nauseam. Mathematics cannot be proven to 'exist' in the Platonic sense, but the idea that mathematics reveals fundamental truths about the reality in which we live is a useful one. Simply citing Quantum Mechanics and saying "See? It's non-deterministic, therefore deterministic systems do not reflect reality" simply does not follow, since after all, logic is also a deterministic system. If you deny logic, then why bother making an argument?

It is possible that our universe is inherently paradoxical and illogical, which would seem to be the logical end to your argument (or should I say the illogical end?). This is actually one of the central tenets of Zen Buddhism. But Zen requires satori, and I'm not enlightened so I cannot say what Zen reveals about the nature of ourselves and the universe. Sufism and Jewish Kabbalah also have the notion of 'higher truth' which is not bound by the rules of logic. But they also state explicitly that discussion of such truths is useless, since this 'truth' is hidden and cannot be communicated with language (much like you cannot explain to someone who has been blind all their life what it's like to see). But I digress.

Also, I take issue with your use of the term 'arbitrary'. Arbitrary does not mean "random" as you seem to imply. It means specifically chosen by someone. So yes, the axioms of mathematics are arbitrary, but they are chosen such that they 'make sense', and so far this has been very useful for understanding our universe. As I pointed out before, you can't take quantum mechanics seriously unless you take its mathematics seriously, since (as you know) it's highly abstract and much of it only makes sense in mathematics, though its results are supported by experiment.

So, is mathematics 'real' in the Platonic sense? That's metaphysics, and really beside the point. Does mathematics currently provide insight into the nature of the universe, and will it continue to do so in the future? In my opinion, definitely. Does this connection have any 'higher meaning', such as the existence of a creator, or the concept of the harmonious interconnectedness of all things? Despite the length of this post, I must say in all honesty, I don't know. :)

Inventors (none / 0) (#49)
by kinkie on Tue Jan 09, 2001 at 05:00:50 AM EST

For a very simple reason: maths is based on axioms, and the only reason we are using THIS math and not another is that the current set of axioms and the math that can be derived from them fit well to the purpose of explaining the physics of the world as we see them
It might very well happen that at some point some physics discovery will be made that will require a different kind of math to be derived from different axioms.

The point is, those axioms are arbitrary in nature, and thus are "invented", not "discovered". This IMO makes the difference.

/kinkie
Except that... (none / 0) (#52)
by dcturner on Tue Dec 17, 2002 at 06:58:55 PM EST

Firstly, a big pile of maths was discovered/invented before axiomatic set theory came about, then Zermalo and Fraenkel reverse engineered their axioms to fit. So you might say they discovered them as a fundamental description of the rest of maths at the time.

Secondly, once you've got the axioms of ZF, is everything that follows also an invention or is it then a discovery?

Is there a invention/discovery duality akin to the wave/particle duality? I say yes.

Remove the opinion on spam to reply.


[ Parent ]
Absurd questions requires absurd answers (none / 0) (#50)
by Steeltoe on Tue Jan 09, 2001 at 09:58:21 AM EST

This question, like so many other, is absurd. Simply because it heavily depends on how you experience, define and understand the universe. No matter how much you think about it or discuss it with peers you'll come to four general models of the universe/self:

1) The universe is a static multi-dimensional picture. Everything is deterministic. So our process as human beings is merely a tiny facet of this, and the "time" we experience nothing else than a trajectory throughout this multi-dimensional canvas.

2) The universe is a dynamic multi-dimensional process. The next step in this process has an undeterministic quality on the low (quantuum) levels, so time really exists along with space, creating a process. Not perhaps our kind of "time", but something similar on a higher level.

3) The infinite universe. This model effectively combines 1) and 2), but may ONLY be discussed at a certain Restaurant frequently visited by galactic hitchhikers. There's just no other hope in making an accurate infinite model of an infinite reality.

4) The universe just ISN'T. Another absurd claim which may be closer to the truth than most will admit, even to themselves. How can we admit it, if when we fully accept this, we simply cease to ever have been?

So, to sum it up: This is all pretty absurd. Define what words you like, and put them in neat little contexts. If they work, use them.

I doubt we can ever have any hope of fully explaining existance or generalize what we experience. We can only BE. Any language trying to explain it will simply be too limited, related and defined to encompass that. However, all models are useful at the time and place they're needed. :-)

- Steeltoe
Explore the Art of Living

What is Mathematics Really? (none / 0) (#51)
by danny on Thu Jan 11, 2001 at 11:05:32 PM EST

There's a great book by Reuben Hersch called What is Mathematics, Really?, that I recommend to anyone interested in the philosophy of mathematics. It's on my list of books to review...

Danny.
[900 book reviews and other stuff]

Is mathematics invention or discovery? | 52 comments (48 topical, 4 editorial, 0 hidden)
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