Was math discovered or invented?

[alterego]

I was just sitting around and pondering this and I cant choose one or the other, though I do kind of lean toward invented.

What do you think? Was math discovered or invented and why?

Erase man and you erase math right?

[biqu]

The answer probably lies somewhere in between the two extremes, and the exact location on that scale might not have a factual answer.

That's why this previous thread on the subject (http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=68600) was found in GD rather than GQ.

[alterego]

I guess my search (http://boards.straightdope.com/sdmb/search.php?action=showresults&searchid=865477&sortby=lastpost&sortorder=descending) was a little frugal being as it only lists my thread in the results =)

~ Thanks ~

[alterego]

I have to say though that the search function on this board seems to be a little lacking. Had I entered 'math discovered invented' into Google which I normally use for searching that thread would have come up.

[biqu]

Originally posted by alterego
I have to say though that the search function on this board seems to be a little lacking. Had I entered 'math discovered invented' into Google which I normally use for searching that thread would have come up.

What were the keywords you used in your search? What time interval did you specify? Which forums did you search? Neither Google nor the SDMB search engine can read your mind (http://slashdot.org/articles/03/07/22/1411240.shtml?tid=126&tid=95), so your search must be refined until it produces the results you want.

Since this post so far adds nothing to the discussion, I'll make some redeeming comments now. I'll quote from the mathematician/philosopher Gian-Carlo Rota, among whose essays you'll find plenty of discussion on this issue. Since he died in 1999 (http://web.mit.edu/newsoffice/nr/1999/rota.html), his work is not yet public domain, so you'll have to trek to your local library if you want to read the essays in their entirety.

Are mathematical ideas invented or discovered? This question has been repeatedly posed by philosophers through the ages and will probably be with us forever. We will not be concerned with the answer. What matters is that by asking the question, we acknowledge that mathematics has been leading a double life.

In the first of its lives mathematics deals with facts, like any other science. It is a fact that the altitudes of a triangle meet at a point; it is a fact that there are only seventeen kinds of symmetry in the plane; it is a fact that there are only five non-linear differential equations with fixed singularities; it is a fact that every finite group of odd order is solvable. The work of a mathematician consists of dealing with such facts in various ways. <snip>

In its second life, mathematics deals with proofs. A mathematical theory begins with definitions and derives its results from clearly agreed-upon rules of inference. Every fact of mathematics must be ensconced in an axiomatic theory and formally proved if it is to be accepted as true. Axiomatic exposition is indispensable in mathematics because the facts of mathematics, unlike the facts of physics, are not amenable to experimental verification.

The axiomatic method of mathematics is one of the great achievements of our culture. However, it is only a method. Whereas the facts of mathematics once discovered will never change, the method by which these facts are verified has changed many times in the past, and it would be foolhardy to expect that changes will not occur again at some future date. <snip>

The facts of mathematics are verified and presented by the axiomatic method. One must guard, however, against confusing the presentation of mathematics with the content of mathematics. An axiomatic presentation of a mathematical fact differs from the fact that is being presented as medicine differs from food. It is true that this particular medicine is necessary to keep mathematicians from self-delusions of the mind. Nonetheless, understanding mathematics means being able to forget the medicine and enjoy the food. <snip>

Nowhere in the sciences does one find as wide a gap as that between the written version of a mathematical result and the discourse that is required in order to understand the same result. The axiomatic method of presentation of mathematics has reached a pinnacle of fanaticism in our time. <snip> This pretense of "identifying" mathematics with a style of exposition is having a corrosive effect on the way mathematics is viewed by scientists in other disciplines. <snip> The mistaken identification of mathematics with the axiomatic method has led to a widespread prejudice among scientists that mathematics is nothing but a pedantic grammar, suitable only for belaboring the obvious and for producing marginal counterexamples to useful facts that are by and large true.

[biqu]

Originally posted by amore ac studio
Since he died in 1999, his work is not yet public domain, so you'll have to trek to your local library if you want to read the essays in their entirety.

As far as I can tell, the essays from which I quoted were originally written in English, since no translator was credited on the title page. I'm not sure what efforts a library in Naples, Italy would make to acquire foreign-language titles. On the other hand, if you can read Italian, you could probably find an Italian translation of Rota's works more easily than the original English version.

[Hari Seldon]

Yes, the matter has been debated at length, to no conclusion. I can tell you only of my own experience which is that most of the time I feel that I have made a discovery of something that was there, just waiting to be found and I was the first one to find it. But once, just once, in my 40 year career, I did something that truly felt like an invention. I guess there have been a couple other times that I did something that seemed like at least a semi-invention. But with the one exception noted above, I always felt that I was following in someone else's footsteps, just going on bit further.

Actually, I think the question is not well-phrased since I am not sure the terms have clear-cut meanings.

[Phage]

Math is similar in origin to logic itself. If you have one item, and then you get another single item, two stands for how many items you have. Obviously the linguistic representation of numbers was invented by humans, but it was invented to describe how things operate in the world. No one was walking along a beach one day and found a number in the sand, it is an idea.

Just because it is an idea, however, does not mean that by destroying those who know of the idea it will be destroyed. The language would be different, but any intelligent creature can see that one and two are distinct concepts. They are as much an inherent property of the universe as our 3-dimensional world, and although the current theories and terminology might be lost it will not change.

The difficulty arises in deciding if describing an inherent property is a discovery, or a cognitive invention. On one hand you can argue that we did not create the universe’s properties, and therefore our description of them is a discovery of things that are. If you think like this then you can also extend that thinking to your own body and mind; you did not make them, so you discovered them. On the other hand, you can choose to consider math and all other descriptive theories as inventions of humans, and while the same conclusions can be arrived at by other people the theories are still original works.

With both hands, you might make some sense out of it.

[DrMatrix]

Off to Great Debates.

DrMatrix - GQ Moderator

[ultrafilter]

I have no opinion on the subject. A mathematician's belief on this matter doesn't make a bit of difference in his ability to do math, so I don't see much point in worrying about it.

[green_dragon]

I think of maths as being this abstract universe, with it's own strict laws for what you can or cannot do..mathematicians play around within these rules to try and get to places never been before, find previously unknown connections etc.

So was this universe discovered or invented? How about the rules which govern it? I think the rules are abstract representations of logical thought, a device to aid the mind in the process of deduction. If that were the case, I would lean towards them being discovered - after all, there's really no other way to express them other than the way they are.

Seeing as the whole shebang is built on rules I say discovered

However, the notations, language etc. were invented.

[FairDink]

Thats right! Language was invented. But what was it invented for? For the same reasons that maths was invented... Communication. Communication of thought and emotions have been one of the most (if not the most) active area of human evolution. Many animals can feel happy, sad etc, but to 'think it' is a different story. The ability to 'think it' suddenly requires the ability to comunicate it to others. As humans tend to think way too much, we also tend to 'invent' more ways to communicate what we think (just look around this cite). Now, thought comes in many different flavours. One of them being based on observations and trying to make predictions based on those observations. My opinion is that maths is one of these 'thought processes'.
Observation is not made up, but the way you wish to describe it to others is completely made up, and the way you would do it depends on your intention for the 'listner'.

[biqu]

Originally posted by Phage
Just because it is an idea, however, does not mean that by destroying those who know of the idea it will be destroyed.

This observation was mentioned in the concluding pages of Friedrich D&uuml;rrenmatt's comedy Die Physiker.

Er versuchte zu verschweigen, was nicht verschwiegen werden konnte. Denn was ihm offenbart worden war, ist kein Geheimnis. Weil es denkbar ist. Alles Denkbare wird einmal gedacht. Jetzt oder in der Zukunft.

He (M&ouml;bius) tried to keep secret that which could not be kept secret. For what was revealed to him is no secret, because it is thinkable. Everything thinkable will one day be thought. Now or in the future.

Originally posted by FairDink
That's right! Language was invented. But what was it invented for? For the same reasons that maths was invented... Communication.

You seem to be equating maths with the notation used to describe mathematical facts, or worse, identifing maths with a style of exposition, as Gian-Carlo Rota warned us against. But maths encompasses more than notation and the axiomatic method. Sure, those are weighty contributions maths has made to modern society, but the idealised picture of what mathematicians do is hardly accurate. It is said that a mathematician is a device for turning coffee into theorems, but the method by which these theorems are produced is grossly mischaracterised in the popular perception of mathematicians' work. Theorems are not, in general, arrived at by strict logical inferences starting with the relevant definitions. On the contrary, proofs of theorems are secondary to the gathering of evidence that suggests which conjectures one can prove.

The following quotes are taken from Jerry Uhl's favourite quotations (http://www.dougshaw.com/uhl/).

The more it approaches intuition, the more reliable the deduction is. Intuition has two distinctive features - it is an instantaneous act and it consists of clear grasp of an idea. Intuition and deduction should be trustworthy processes which we can use to lead to genuine knowledge. There are two ways of arriving at knowledge - through experience and deduction.

-- Rene Descartes

In mathematics, you can be as formal and rigorous as you want to be; but without insight, you'll get nowhere.

-- Richard Feynman

For these two intellectual giants, mathematical facts can be deduced not just from the axiomatic method, but also from a flash of inspiration in which collected pieces of evidence are suddenly seen as examples of a general rule. This experience is viewed by mathematicians in the same way that the discovery of oxygen was viewed by chemists who had grown up believing in phlogiston theory. Our own RTFirefly experienced this feeling (http://boards.straightdope.com/sdmb/showthread.php?postid=1278451#post1278451) in his work on graph theory.

Some defects ordinarily found in the method used by mathematicians:

Defect 1: To pay more attention to proof than to evidence, and to try to convince rather than enlighten the mind.

Defect 2: To prove things that do not require proof.

-- Antoine Arnaud and Pierre Nicole

If mathematics were formally true but in no way enlightening, then mathematics would be a curious game played by weird people. Mathematicians seldom explicitly acknowledge the phenomenon of enlightenment.

-- Gian-Carlo Rota

These comments reflect their authors' belief that mathematical facts are not just the result of applying the rules of logical inference to an established set of definitions. On the contrary, mathematical facts reflect truths about the world. These truths can be inferred from observations, and they can enlighten the mind with their beauty and generality, lifting one momentarily into the ideal world of Plato's forms.

[FairDink]

You seem to be equating maths with the notation used to describe mathematical facts, or worse, identifing maths with a style of exposition, as Gian-Carlo Rota warned us against.


On the contrary, I was trying to highlight that mathematics refers to many aspects of a whole - not just one aspect of it. It is the collective term used to describe, among other things its notation. If a person was to ask me "OK, lets talk mathematics!". How else can we do this without refering to its notation. So language is an important part of mathematics, but i'm not suggesting that it IS mathematics - far from it if you read it carefully. I tried to show three distinctive components - obviously I failed, but here they are: Existence, Observation, Communication.
One cannot refer to one without using another.
Now, the human brain finds it easier to work on defined principles, that is where the communication comes in. Even if there were only one person on this planet, He/She would find easier to build on ideas if they could write them down, or at least name/define/objectify them.

May I also add that 'observation' and 'prediction' as mentioned in my previous context need not necessarily be a concious effort.


For these two intellectual giants, mathematical facts can be deduced not just from the axiomatic method, but also from a flash of inspiration in which collected pieces of evidence are suddenly seen as examples of a general rule.


A 'collection' of many subconcios observations may be a key in piecing together two or more concious ones, giving a kind of 'realization sensation'.

[biqu]

Originally posted by FairDink
Many animals can feel happy, sad etc, but to 'think it' is a different story. The ability to 'think it' suddenly requires the ability to communicate it to others. As humans tend to think way too much, we also tend to 'invent' more ways to communicate what we think (just look around this cite).

I fail to see why the ability to think a thought requires the ability to verbally communicate it to others. Here's what Einstein had to say about thinking and verbalization.

I insist that words are totally absent from my mind when I really think . . .

. . . Even after reading or hearing a question, every word disappears at the very moment I am beginning to think it over; words do not reappear in my consciousness before I have accomplished or given up the research.

. . . and I fully agree with Schopenhauer when he writes, Thoughts die the moment they are embodied by words. I think it is also essential to emphasize that I behave in this way not only about words, but even about algebraic signs. I use them when dealing with easy calculations; but whenever the matter looks more difficult, they become too heavy baggage for me.

There was a science fiction novel Neanderthal that some reviewers said would become the next Jurassic Park in terms of capturing the mindshare of readers. The story tells of a civilization of Neanderthals that lives in a high mountain range and has escaped human notice since their respective civilizations diverged. The anthropologists and scientists who observe the Neanderthals find it surprising that they have not developed a language to facilitate communication. As it turns out, they didn't require a traditional language, because they had developed a telepathic ability that permitted communication without words.

Based on Einstein's quote and the plausible scenario depicted in the novel Neanderthal, I would conclude that the ability to think a thought does not depend on having a way of expressing that thought in words.

I was trying to highlight that mathematics refers to many aspects of a whole - not just one aspect of it. It is the collective term used to describe, among other things its notation. If a person was to ask me "OK, lets talk mathematics!". How else can we do this without refering to its notation?

When I quoted you as saying that "maths was invented . . ." to facilitate communication, it seemed you were attributing the property of being invented to all of mathematics, not just the notation or the vocabulary needed to communicate mathematical ideas. If that was your intent, then you are also saying that Existence and Observation (the two other components of mathematics you listed) were also invented, which I have a hard time accepting. I agree that the notation and vocabulary of mathematics were invented, just as English or any other language was invented. But just as words in English refer to objects accessible to our senses (vision, hearing, smell, taste, touch, and also thought), so too does the invented mathematical vocabulary refer to objects accessible to the faculty of thought. This assertion is hard to fathom for Cartesian dualists, who hold that mind is separate from, and of a different essence than, matter. For disciples of eastern religious traditions, especially Buddhism, it is the opposite assertion that is hard to fathom. Sensations, perceptions, consciousness, and thought all belong to the five aggregates (pancakkhandha), and as such are conditioned states, arising out of contact with the physical world and the world of ideas. See for example the exposition in Walpola Rahula's What the Buddha Taught, chapter 2 (The First Noble Truth: Dukkha).

[SentientMeat]

Originally by Albert Einstein, 1921
Insofar as the expressions of mathematics refer to reality they are not certain, and insofar as they are certain they do not refer to reality. Never has there been a more concise expression of the relationship between maths and “the world”.

[blowero]

Originally posted by amore ac studio
There was a science fiction novel Neanderthal that some reviewers said would become the next Jurassic Park in terms of capturing the mindshare of readers. The story tells of a civilization of Neanderthals that lives in a high mountain range and has escaped human notice since their respective civilizations diverged. The anthropologists and scientists who observe the Neanderthals find it surprising that they have not developed a language to facilitate communication. As it turns out, they didn't require a traditional language, because they had developed a telepathic ability that permitted communication without words.

Based on Einstein's quote and the plausible scenario depicted in the novel Neanderthal, I would conclude that the ability to think a thought does not depend on having a way of expressing that thought in words.

Plausible?

[FairDink]

amore ac studio... But if would like someone else (who could not read minds) to see what you have seen then what???


they had developed a telepathic ability that permitted communication without words

Could it have been very sublimininal body language instead???
Or did the scientists look into that?

I agree with you that my verbalization in my original post... 'maths was invented...' was incorrect. However, I still beleive that:
Thinking + Friends = Communication. (not necessarily language).

... And observation can also be considered as interpretation, therefore it is somewhere inbetween reality and made up. For instance, we may interpret an event as happening before some other event, but thaks to our friend Einsteine, we know that it is all relative.
It can also be argued that Albert may not have found his General relativity as easy to 'picture' if he did not define his Special theory so intricately. That is not to say he wouldn't have, but actually defining something (mentally or linguistically) anables the brain to eaither build on "it", or to break "it" down further.

[FairDink]

On the other hand, we may know (at least strongly assume) that there 'exists' a Grand Unified Theory, but we cannot observe it. But that is not to say it doesn't exist.

Look, I will not argue with Einstein, Schopenhauer or you about verbalizing a thought. I agree with you. But to be able to refer to something mentally, even mentally refering to something as "it", is defining something. And defining something to ones self is self-communication.

Actually, come to think of it, "it" is probably the most accurate 'word' you could use to describe everything. But you would not get very far unless we could harness the power of the above mentioned Neanderthals. I am not saying that mental telepathy is not possible, I would actually argue the opposite (in another thread perhaps), what I am trying to say is that by defining something, you have 'something' to communicate. You could share it with others (if they understand), or you can keep it as a mental 'bookmark' for only you to use (self-communicate).

[biqu]

Originally posted by blowero
Plausible?

Not on the planet we know. Perhaps a better phrasing would have been, "not beyond the realm of possibility." Who knows? If life on the planet had developed in a different way, the situation described in the novel could have occurred.

Originally posted by FairDink
Could it have been very sublimininal body language instead???
Or did the scientists look into that?

If you haven't read the novel yet, but are considering it, I won't spoil it for you any more. Just contain your curiosity and don't highlight the spoiler box below.

As it is explained in the story, the Neanderthals had the ability of telepathy with others of the same species, but with humans they were only able to exercise the reduced functionality of "remote viewing", i.e., perceiving in their minds what humans in the vicinity were looking at. These abilities drop off with distance, but they do not depend on visual contact with the person whose mind or vision is being probed.

But if you would like someone else (who could not read minds) to see what you have seen; then what??? <snip>

I agree with you that my verbalization in my original post... 'maths was invented...' was incorrect. However, I still beleive that:
Thinking + Friends = Communication. (not necessarily language).

In Ayn Rand's novel The Fountainhead, the protagonist Howard Roark argues that he cannot perform the act of thinking for another person. All he can do is provide input for another person, which will be digested by that person's mental faculties in a unique way. In a less provocative context, Richard Benson makes much the same point:

There's a terrible problem that I run into in teaching, which is that when you tell people something, you keep them from knowing it. If they find it on their own, they'll know it in a way they never will if you tell them. What I try to do more and more is to bring students to my studio and get them really working.

Of course, none of these quotes disagrees with the point you are making, but they do put restrictions on the extent to which language (even the language of mathematics) can convey something as complex as a thought.

[Deep Throat]

Hmm, to make it simple...

Math is to discovered as Calculus is to invented!

:smack:

[FairDink]

Thanks for the I will read the novel... Eventually.

Hey, I just 'realized', according to my formula [above], making friends is easy:

Communication - Thinking = Friends!!!

[FairDink]

Thanks for the recomendation, that is.

[FairDink]

Sorry about having three posts in a row (maybe it could be a title for a new thread: Is FairDink trying to increase his posts).

But I guess the conclusion is (correct me if i'm wrong) that it depends on which part of mathematics you are refering to?!?!?

[biqu]

Originally posted by FairDink
Thanks for the recommendation. I will read the novel... Eventually.

Here's the Amazon.com listing (http://www.amazon.com/exec/obidos/ASIN/0312963009/qid=1063076498/sr=2-1/ref=sr_2_1/102-2590510-6201719) for John Darnton's Neanderthal.

But I guess the conclusion is (correct me if i'm wrong) that it depends on which part of mathematics you are refering to?!?!?

In a nutshell, yes. That's why my first post to this thread stated that neither of the two extremes of "discovered" or "invented" accurately captured the nature of mathematics.