Math discovered or invented?

For the discussion of math. Duh.

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heyitsguay
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Re: Math discovered or invented?

Yakk wrote:The thing that is created when we do mathematics is the proof that connects axiom to theorem. The space of "true statements" that are implied even by axioms (and the conceptual framework that lets you prove things with them) is vast, and speaking of the consequences as being "already there, just not discovered yet" is questionable.

It is like saying that all books in existence are implied by the existence of a printing press, and any book after that point was merely a discovery of the arrangement of characters. Or if you dislike that, the existence of number based encoding of books (where a book is represented by a single number, and all possible books are listed in this index uniquely) made the act of writing one of discovery of those numbers.

There is one critical difference that your analogy passes over however: The semantic content of books is not a function of the semantic content of letters, the characters which comprise the books. In math on the other hand, the opposite of this is true. Once a few axioms and their symbolic representations are agreed upon, the exact meaning of a mathematical statement can be read off (at least in theory) from the arrangements of these symbols, even though in practice that's not quite how things are done. Just knowing what "c", "a", and "t" are does not tell you what "cat" means as an English word, but knowing what [imath]P,Q,\Rightarrow[/imath] mean allows you to deduce the meaning of [imath]P\Rightarrow Q[/imath].

bitsplit
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Re: Math discovered or invented?

heyitsguay wrote:
Yakk wrote:Just knowing what "c", "a", and "t" are does not tell you what "cat" means as an English word, but knowing what [imath]P,Q,\Rightarrow[/imath] mean allows you to deduce the meaning of [imath]P\Rightarrow Q[/imath].

I think your point loses a bit if strength if you use words instead of letters as the base symbols of language. I mean, knowing what cat, dog, and chases means in English tells you something about dog chases cat. There are other languages where a single symbol represents an entire idea. Mathematics uses the same concept, having many symbols, instead of a smaller number of symbols (letters) that lump together to represent the elements of an expression.

It could be argued, as well, that you can't just throw a random array of words in English and call it a sentence. But in the same manner, you can't just throw a random array of mathematical symbols and call it a well formed expression.

I do, however, believe there is much more constraint in proof than in English. But it follows from logical context. It has more to do with the limitations of the content. Mathematical proofs bind you very tightly to what you can talk about. Particularly, what you write has to follow from what has been shown before, has to be a definition, or has to be an assumption and stated as such. Human languages, in general, allow much more leeway in their subject matter. You can talk about your mother, and then go on a rant about flying unicorns from space that ate your car. Maybe nobody would care, maybe they'd think you insane, but it would be valid language nonetheless.

That does not mean, however, that there is no inventiveness in proof. Most proof stems from a process that is not really logical at all. People don't think like machines. The logic flows from a rational process that gives rigorous form to understanding. There is stumbling about, there is rethinking. Well, maybe I am wrong, but that is the way I have usually seen it done, and seen it discussed. I think mathematicians are some of the most creative and imaginative people I have ever come across. They have the most vivid and abstract imaginations I have come across. They discover knowledge, and create ways to manipulate that knowledge, that is seemingly inconceivable to most other human beings. And to top it all, they take all that imagination, and creativity, and crystallize it into something as concrete as a proof. Once it's there, it's there, and there's no turning back. The truth exists outside their imagination, and their creation; it is a separate thing. The proof is a child of that process, and the truth is what lies beyond the door it opened.

Ok, I find that my posts are overly long, and I've posted entirely too many times on this thread. I'll shut up now. I think I've said everything I've had to say.

Cmebeh
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Re: Math discovered or invented?

Is it safe to say the axioms are invented and the theorems and proofs are discovered?

I dont think natural numbers exist outside of human experience (or the experiences other highly intelligent beings that can define numbers). Do other forms of life experience the natural numbers? What about jelly fish? Think of what a jelly fish experiences and tell me what stimuli it processed to deduce the concept of natural number. I think the answer to the natural number question lies in cognitive science. The evidence i feel against natural numbers existing outside human experience is:
0) Zero wasn't invented for a long time.
1) The axiom of infinity is independent of the others.
2) not all lifeforms need numbers to survive
3) The cardinal number of a set is independent of what actually constitutes the set, ie |{a,b,c}| = |{cat, dog, {1,2,3}}|. Numbers are a relation between sets, and relations are defined by humans. For example, you ask for 2 slices of pizza. the pizza guy give you 1 slice. you say hey i asked for 2 slices. so he takes that one slice and cuts it in half and gives them to you. So do you have 2 slices of pizza or do you have 2 half slices of pizza? Well maybe the first slice was the size of 2 slices, so you didn't start with 'one' to begin with. Its all qualitative, and the qualities are defined by the people involved, so there is no such thing as number and im not paying for the pizza.
Cmebeh wrote:LOL you computer scientists; isn't it obvious? P = NP is undecidable à la continuum hypothesis

Cleverbeans
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Re: Math discovered or invented?

Cmebeh wrote:Is it safe to say the axioms are invented and the theorems and proofs are discovered?

I dunno, the standard for what constitutes an acceptable proof is a social phenomenon, I'm not sure it exists outside the human experience.
"Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration." - Abraham Lincoln

achan1058
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Re: Math discovered or invented?

Cleverbeans wrote:
Cmebeh wrote:Is it safe to say the axioms are invented and the theorems and proofs are discovered?

I dunno, the standard for what constitutes an acceptable proof is a social phenomenon, I'm not sure it exists outside the human experience.
We can always reduce everything to the formal string games, which are verifiable by computers and etc. So in theory we can say it is "discovered", once a sufficiently formal theory is invented.

supremum
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Re: Math discovered or invented?

achan1058 wrote:
Cleverbeans wrote:
Cmebeh wrote:Is it safe to say the axioms are invented and the theorems and proofs are discovered?

I dunno, the standard for what constitutes an acceptable proof is a social phenomenon, I'm not sure it exists outside the human experience.
We can always reduce everything to the formal string games, which are verifiable by computers and etc. So in theory we can say it is "discovered", once a sufficiently formal theory is invented.

Umm...wouldn't there still be metamathematical statements that wouldn't be provable in that theory?

Cleverbeans
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Re: Math discovered or invented?

achan1058 wrote:We can always reduce everything to the formal string games, which are verifiable by computers and etc.

Oh? And what proofs are you basing that on exactly?
"Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration." - Abraham Lincoln

achan1058
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Re: Math discovered or invented?

Cleverbeans wrote:
achan1058 wrote:We can always reduce everything to the formal string games, which are verifiable by computers and etc.

Oh? And what proofs are you basing that on exactly?
That's what axiomatic set theory is. Reducing to this level will ensure that everything that is socially dependent is seated in the axioms (including the axioms of logic, of course).
supremum wrote:Umm...wouldn't there still be metamathematical statements that wouldn't be provable in that theory?
You mean statements such that it and its negations are not provable? Yes, there would be, but does it matter?

Torn Apart By Dingos
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Re: Math discovered or invented?

Cmebeh wrote:Is it safe to say the axioms are invented and the theorems and proofs are discovered?

I dont think natural numbers exist outside of human experience (or the experiences other highly intelligent beings that can define numbers). Do other forms of life experience the natural numbers? What about jelly fish? Think of what a jelly fish experiences and tell me what stimuli it processed to deduce the concept of natural number. I think the answer to the natural number question lies in cognitive science. The evidence i feel against natural numbers existing outside human experience is:
0) Zero wasn't invented for a long time.
1) The axiom of infinity is independent of the others.
2) not all lifeforms need numbers to survive
3) The cardinal number of a set is independent of what actually constitutes the set, ie |{a,b,c}| = |{cat, dog, {1,2,3}}|. Numbers are a relation between sets, and relations are defined by humans. For example, you ask for 2 slices of pizza. the pizza guy give you 1 slice. you say hey i asked for 2 slices. so he takes that one slice and cuts it in half and gives them to you. So do you have 2 slices of pizza or do you have 2 half slices of pizza? Well maybe the first slice was the size of 2 slices, so you didn't start with 'one' to begin with. Its all qualitative, and the qualities are defined by the people involved, so there is no such thing as number and im not paying for the pizza.

I agree with your first sentence but I'm not sure I agree with your statement about integers. You've heard "God made the integers; all else is the work of man", right? The axioms and definitions for everything else are somewhat arbitrary and could be defined subtly differently (an alien race who needed geometry might never have looked at Euclidean geometry), but integers seem so natural (if you'll excuse the pun) that it's hard to imagine there could be intelligent lifeforms which don't understand them. Certainly a jellyfish can see one other jellyfish, or two, or several.

A fun thought experiment is to try to imagine some alien race whose first proof for the infinitude of the primes was Furstenberg's topological proof. Maybe they live in some kind of world with no obvious separation of the world into discrete objects. Maybe it's a single giant gelatinous blob of consciousness which lacks vision. Certainly continuity would be an important concept to it, and thus, it might devise a theory of topology. Then, during mathematical research, it stumbles upon this strange object: an integer, and then uses topological methods to prove that there are infinitely many primes.

I think that's a fun thought, and as much as I'd like to believe it, I'm inclined to believe that the concept of integers is something that any intelligent lifeform would discover very early.

Yakk
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Re: Math discovered or invented?

An incompressible universe wouldn't have anything corresponding to the integers, I suspect.

Because you'd be able to describe two things as being two of the same kind of thing, which is a huge amount of compression. Either that "there are two things" statement isn't all that useful (with the two things in question being so different that this doesn't reduce their collective description by all that much), or it is meaningless.

Of course, an incompressible universe (read: max entropy universe, and not just a local max!) is a very strange beast. I doubt anything resembling human intelligence could exist there.

So long as there are things that are sufficiently similar to all be called "things" and that being useful, counting ends up being powerful -- it is Run Length Compression, in a sense, followed by deltas from the thing you are using as the archetype. The fact that RLC is a very simple and effective form of compression in a universe where things tend to be similar from place to place (so, pebbles tend to be not-completely-unique, and even the concept of thing helps grasp what is going on from place to place) means that the integers are really handy and all over the place...

...

The jokes about "and how would you prove that" seem to be standard infinite descent of "mechanical proof" jokes, right?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Tibixe
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Re: Math discovered or invented?

Mathematics is neither discovered nor invented. Nature is coerced by proofs to obey theorems.

Mathematics is an example of our will to power. It is a process, a war against uncertainty in Nature.

( from Gödel's results it is obvious that as long as we base our knowledge on recursively enumerable systems of axioms, no final victory is possible in this war )

BlackSails
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Re: Math discovered or invented?

Tibixe wrote:Mathematics is neither discovered nor invented. Nature is coerced by proofs to obey theorems.

Mathematics is an example of our will to power. It is a process, a war against uncertainty in Nature.

( from Gödel's results it is obvious that as long as we base our knowledge on recursively enumerable systems of axioms, no final victory is possible in this war )

I cant tell if you are joking.

Yakk
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Re: Math discovered or invented?

I think it might be a white wolf mage technocracy joke?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

bitsplit
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Re: Math discovered or invented?

Tibixe wrote:Mathematics is neither discovered nor invented. Nature is coerced by proofs to obey theorems.

Mathematics is an example of our will to power. It is a process, a war against uncertainty in Nature.

( from Gödel's results it is obvious that as long as we base our knowledge on recursively enumerable systems of axioms, no final victory is possible in this war )

Thanks. I needed a good laugh.

black_hat_guy
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Re: Math discovered or invented?

Math was essentially created by humans. A lot of physics (quantum mechanics, relativity, string theory ) does not follow "standard mathematics" Math is based off of what we have observed the Universe to be, and yet is still useful for figuring stuff out. Plus, not only did we invent the symbols, but we also invented the meanings of the symbols. It would probably be possible to represent mathematics using symbols with different meanings.

supremum
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Re: Math discovered or invented?

black_hat_guy wrote:Math is based off of what we have observed the Universe to be.

I disagree. There is lots of perfectly valid math out there that doesn't obviously correspond to anything in the universe at all.

Cleverbeans
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Re: Math discovered or invented?

supremum wrote:I disagree. There is lots of perfectly valid math out there that doesn't obviously correspond to anything in the universe at all.

False, math is in the universe, and there is no perfectly valid math that doesn't relate to other math.
"Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration." - Abraham Lincoln

bspus
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Re: Math discovered or invented?

Would you consider imaginary numbers to be an invention?

if the sqrt of -1 doesn't exist, yet we use it, how can we say that any "proof" we derive using it is a truth that somehow relates to this world?

My knowledge of imaginary numbers is very limited, so I might have completely missed the point, but since nobody mentioned this up to now I thought I should

WarDaft
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Re: Math discovered or invented?

bspus wrote:Would you consider imaginary numbers to be an invention?

if the sqrt of -1 doesn't exist, yet we use it, how can we say that any "proof" we derive using it is a truth that somehow relates to this world?

My knowledge of imaginary numbers is very limited, so I might have completely missed the point, but since nobody mentioned this up to now I thought I should

Imaginary numbers actually have practical real world use.

As does the sum of all positive integers.
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Xanthir
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Re: Math discovered or invented?

Further, the square root of -1 exists precisely as much as the square root of 1.
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))

supremum
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Re: Math discovered or invented?

Cleverbeans wrote:
supremum wrote:I disagree. There is lots of perfectly valid math out there that doesn't obviously correspond to anything in the universe at all.

False, math is in the universe, and there is no perfectly valid math that doesn't relate to other math.

Oops, sorry, bad word choice on my part. When I said "obviously," I meant "directly."

BlackSails
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Re: Math discovered or invented?

bspus wrote:Would you consider imaginary numbers to be an invention?

if the sqrt of -1 doesn't exist, yet we use it, how can we say that any "proof" we derive using it is a truth that somehow relates to this world?

My knowledge of imaginary numbers is very limited, so I might have completely missed the point, but since nobody mentioned this up to now I thought I should

The sqrt of -1 exists in the same sense that -1, 1 and 6 all exist.

edahl
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Re: Math discovered or invented?

BlackSails wrote:
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.

Nice point (assuming space is smooth and complete, I suppose.)

EDIT: I'd like to add that I don't think mathematics is either or. I'd consider imaginary numbers an invention, I'd consider the theory behind the quotient [imath]R[X]/(X^2+1)[/imath] an invention, and I'd consider the isomorphism [imath]C\approx R[X]/(X^2+1)[/imath] a discovery.

vookaloop
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Re: Math discovered or invented?

Argency wrote:
Let A be the set of axioms underlying modern mathematics - the semantic and syntactic rules we employ when we do maths and everything underlying them. Let T be the set of truths which we are able to establish based on these axioms. So, when we do maths, we are asserting that

A ⊃ T

(Ie, the truth of the axioms implies the truth of the truths) What you're saying is that T is false if A is false. This means that

T ⊃ A

(The truth of the truths implies the truth of the axioms or, to put it another way, the falsity of the axioms implies the falsity of the truths) I agree. These two taken together dictate that...

A ≡ T

(The truth of the axioms is logically identical to the truth of the truths) And I agree with all that. All I'm saying is that the above statements are correct whether or not we hold the mathematical axioms to be true. That is, whether A and T are true or false, the symbols ≡ and ⊃ always have truth value 1 in the above statements. This is something which we have discovered on examining our invented mathematical axioms.

I just started studying symbolic logic this quarter, but to my understanding, "if T is false then A is false" would not be accurately represented by "T ⊃ A ", because a false antecedent implies a true consequent. Am I wrong?

Kurushimi
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Re: Math discovered or invented?

He said "T is false if A is false", or "if A is false, T is false", so he is correct on that point.

YamaNeko
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Re: Math discovered or invented?

My question to you all is, what do you consider mathematics?
Hard to discuss about something so vague as mathematics before it is well-defined don't you think?

To me, Mathematics is the use of classical logic to form conclusions or gain knowledge of further truth.
This sounds very similar to philosophy, however, the difference with philosophy and mathematics is our axiomatic set theory, where we claim that these are fundamentally true.

Furthermore:
Discovery implies that something has always existed, while invention implies that it has not always existed.

With this new word, "exist", we get into very philosophical territory about what it means to have existed or exist. Since this is not a thread on existentialism, I'll let you guys decide on whether or not something is considered or not to be "existing".

---------------------

Now, with the definitions of "Discovery", "Invention", and "Mathematics" discussed, I can provide an answer to your question.

If you claim Mathematics is a discovery, you are claiming that our Axiomatic set theory (which will conclusively lead to the fact that the Pythagorean theorem is true, since that was a topic discussed in this thread) has ALWAYS existed.

If you claim Mathematics is an invention, you are claiming that our Axiomatic set theory has NOT ALWAYS existed.

===============================================

Personally, I believe that our Axiomatic Set Theories always existed, therefore it would not be proper to call it an invention since invention implies non-existence, and therefore it is more proper (not necessarily) to call it a discovery.
Last edited by gmalivuk on Thu Aug 05, 2010 10:11 pm UTC, edited 1 time in total.

bitsplit
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Re: Math discovered or invented?

YamaNeko wrote:My question to you all is, what do you consider mathematics?
Hard to discuss about something so vague as mathematics before it is well-defined don't you think?

To me, Mathematics is the use of classical logic to form conclusions or gain knowledge of further truth.
This sounds very similar to philosophy, however, the difference with philosophy and mathematics is our axiomatic set theory, where we claim that these are fundamentally true.

Furthermore:
Discovery implies that something has always existed, while invention implies that it has not always existed.

With this new word, "exist", we get into very philosophical territory about what it means to have existed or exist. Since this is not a thread on existentialism, I'll let you guys decide on whether or not something is considered or not to be "existing".

---------------------

Now, with the definitions of "Discovery", "Invention", and "Mathematics" discussed, I can provide an answer to your question.

If you claim Mathematics is a discovery, you are claiming that our Axiomatic set theory (which will conclusively lead to the fact that the Pythagorean theorem is true, since that was a topic discussed in this thread) has ALWAYS existed.

If you claim Mathematics is an invention, you are claiming that our Axiomatic set theory has NOT ALWAYS existed.

===============================================

Personally, I believe that our Axiomatic Set Theories always existed, therefore it would not be proper to call it an invention since invention implies non-existence, and therefore it is more proper (not necessarily) to call it a discovery.

As always this boils down to nomenclature. I do agree with your approach, where you first define what you call mathematics and then examine whether it is discovered or invented. However, it disregards the probable assumptions others will have, particularly those that would disagree when allowed superficial argument only, about the other definitions the word mathematics might assume.

When you refer to axiomatic set theories, I believe you refer to the truths contained therein and not to the human process of obtaining those truths by stating axioms and deriving theorems through a process of combined intuition, rational thought, random search, and communication with other humans.

The "truths" are discovered, but at least some (or some would say most) of the proofs and methods involving humans regarding those proofs are invented, or evolved. Language, writing, convention, those might be a mixture of evolution and invention. The proofs, at least many of the non-obvious proofs, are invented. They are still mathematical animals of course. But choosing that particular mathematical animal out of the vast sea of possible proofs is a non-trivial act.

Of course, I have no proof of these statements beyond philosophical argument, but they remain my solemn belief, based on my understanding of the mathematical process. To a certain extent, if you believe in determinism, there is no invention or discovery; there merely is or is not. But that is a different argument for a different topic.

King Author
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Re: Math discovered or invented?

@Tirian: For some reason, I bust out laughing at the idea of intelligent dolphins pouring over hydrodynamics. And now I want a snarky t-shirt depicting as much.
I have signitures disabled. If you do, too...you can't read this, so nevermind >_>

RequinB4
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Re: Math discovered or invented?

Relevent quotes:

How did logic come into existence in man's head? Certainly out of illogic, whose realm originally must have been immense. Innumerable beings who made inferences in a way different from ours perished; for all that, their ways might have been truer. Those, for example, who did not know how to find often enough what is "equal" as regards both nourishment and hostile animals--those, in other words, who subsumed things too slowly and cautiously--were favored with a lesser probability of survival than those who guessed immediately upon encountering similar instances that they must be equal. The dominant tendency, however, to treat as equal what is merely similar--an illogical tendency, for nothing is really equal--is what first created any basis for logic.

In order that the concept of substance could originate--which is indispensible for logic although in the strictest sense nothing real corresponds to it--it was likewise necessary that for a long time one did not see or perceive the changes in things. The beings that did not see so precisely had an advantage over those who saw everything "in flux." At bottom, every high degree of caution in making inferences and every skeptical tendency constitute a great danger for life. No living beings would have survived if the opposite tendency--to affirm rather than suspend judgement, to err and make up things rather than wait, to assent rather than negate, to pass judgement rather than be just-- had not been bred to the point where it became extraordinarily strong.

We have arranged for ourselves a world in which we can live - by positing bodies, lines, planes, causes and effects, motion and rest, form and content; without these articles of faith nobody could now endure life. But that does not prove them. Life is no argument. The conditions of life might include error.

The total character of the world, however, is in all eternity chaos--in the sense not of a lack of necessity but a lack of order, arrangement, form, beauty, wisdom, and whatever names there are for our aesthetic anthropomorphisms...Let us beware of attributing to it heartlessness and unreason or their opposites: it is neither perfect nor beautiful, nor noble, nor does it wish to become any of these things; it does not by any means strive to imitate man... Let us beware of saying that there are laws in nature.

Because we have for millenia made moral, aesthetic, religious demands on the world, looked upon it with blind desire, passion or fear, and abandoned ourselves to the bad habits of illogical thinking, this world has gradually become so marvelously variegated, frightful, meaningful, soulful, it has acquired color - but we have been the colorists: it is the human intellect that has made appearances appear and transported its erroneous basic conceptions into things.

~Freidrich Neitzche

silverhammermba
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Re: Math discovered or invented?

I mostly agree with Mario Livio as he puts it in "Is God a Mathematician?" It's a mixture of the two. The axioms and definitions of mathematics are invented: we choose them because they conform to our intuition and because we find them useful in our work. The relationships between these invented objects, however, are discovered.

Obviously if two people can agree on the same set of logical axioms and mathematical definitions then all of mathematics can be objectively determined by both.

Arguing about whether defined objects can exist in an absolute sense strikes me as silly and ultimately impossible. Consider, for example, asking whether a sphere can exist absolutely - independent of human thought. What does that even mean? Without the definition of an object we can't ask if that object exists.

I apologize for disregarding the previous arguments, but my statement does apply. For example, even if we are constructing mathematics simply by applying classical logic to fundamental axioms, classical logic is itself an agreed upon construction. Luckily it is a construction that most people agree with, but it is invented nonetheless. The simplest proof of "P, if P then Q, thus Q" follows rules that are so basic as to be immune to existential arguments.

Jyrki
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Re: Math discovered or invented?

YamaNeko wrote:My question to you all is, what do you consider mathematics?

Ah. The age old question popped up again! No matter how hard you try, the most concise answer to that question ever given is "Mathematics is what mathematicians do". And, no. Don't start complaining about a circular definition, when you press us to define a mathematician.

YamaNeko wrote:Hard to discuss about something so vague as mathematics before it is well-defined don't you think?

No. See the volume of discussion about mathematics around you!

YamaNeko wrote:To me, Mathematics is the use of classical logic to form conclusions or gain knowledge of further truth.

Sounds overly restrictive to me. And also overly broad. For example, mathematicians also do experiments (to gain an understanding of what's going on) before embarking on any attempt at a proof. Sets of axioms evolve. Definitions evolve. And other disciplines also "use classical logic to form conclusions or gain knowledge of further truth".

Back on topic? My take is that parts of mathematics are discovered. Other parts are invented. Sometimes I feel that Math is a science. Sometimes I feel that it is an art. Sometimes I feel that it is a game. This type of discussion is sometimes more fun, if beer and pizza are involved.