Maths

[Paul Doherty]

This led to a big argument in our house last night, aided by two bottles of wine:

- Is Maths a discovery or an invention?

I see (from Google) that it's a much debated point.

[Paul Doherty]

And my only excuse for posting it here is that (a) you're all rather clever and might have a view, and (b) Maths is often said to be "a language". Surely language is an invention? Or is it ...?

Would happiness exist if there was no-one to be happy? Very Zen this afternoon!

[Dave M]

A good one!

I reckon it parallels language: humans made up the words, and the patterns of their use, but what the words are used to express can be the reality already existing around us.

In maths, there are facts relating to numbers and functions, and the STUDY called mathematics has revealed those to us, and helps us to understand and use them. If two shells on a primeval shore were joined by two more, then there'd be four of them - quite unconnected with the derivation by humans of the concept of fourness, the words for "two" and "four", and the concept and effects of addition.

So, we can swing the question back as one about language: does "mathematics" refer to the study and defined knowledge, or to the "mathematical" relationships which are already there?

[Paul Doherty]

Yes, the "two shells plus two shells = four shells" argument was debated last night -- it seems to make a useful distinction between the physical things and the man-man language used to count and describe them. But what about negative numbers -- you can't have "minus six" apples. Or Pythagoras's theorum -- it obviously didn't exist before he invented it, but was it somehow "there" already, lurking in the triangle-ness of triangles, waiting to be discovered.

So what about concepts in general? Do square roots, or the formula for a parabola somehow lurk out there, waiting to be discovered? Can a concept exist if there are no minds to think it? (Thus my "happiness" question.)

And music ... did music exist before man? In birdsong, say. So were Chopin's preludes just waiting for Chopin to come along ...?

[Dr Mildr]

My simplistic view is that it was not invented, but was there and hence has been discovered. When I was a medical lab technician, lots of STIs were 'non-specific'. They are now known to be caused by chlamydia. Chlamydia was always there, but was discovered, not invented. Or perhaps that not the same sort of thing at all.

I'd say an invention is something that results from deliberate thought. It comes about as a result of planning how to solve something. Discoveries are more random. They happen and because they serve a purpose they become reinforced and often become refined (such as language, or counting or whatever). No doubt some (many) discoveries fall by the wayside because they have no use or purpose.

[Paul Doherty]

May 15, 2008 18:29:33 GMT Dr Mildr said:

I'd say an invention is something that results from deliberate thought.It comes about as a result of planning how to solve something. Discoveries are more random.

So take the idea of zero. Man managed without it for a very long time -- it's not an obvious concept -- but was it "out there" waiting to be discovered randomly, or did someone sit down and study the problem of non-positional maths (imagine trying to do mutiplication in roman numerals, or add C to M) and realise they needed positional units and thus a zero symbol?

Seems like an invention to me.

[Sue M-V]

I am sadly ill-equipped to make any pronouncements on mathematics, but I tend to agree with Dr Mildr. I think it's a rationalisation or systematisation of an observable set of facts or truths. I wouldn't say it's a language in itself, although it has developed its own sort of language so that it can be appreciated or absorbed.

Zero is a fascinating concept, and took such a long time for mankind to come up with that it seems more like an invention, but still I wouldn't class it as one. It seems to me to be just an extension of the general idea of absence or lack.

I regard sciences in general as being explanations of things rather than inventions - the latter come rather under the headings of technology or art.

Of course, I'm talking out of the back of my head, since I know nothing of most sciences, including maths.

Sue

[Tone]

There existed (in this current reality) a certain set relationship between things (predominantly quantities).
That was discovered.
Then math was invented to handle that relationship.
That was invented.

Then someone realized that things that didn't actually exist could be handled (usefully) by inventing an extension to math (think Smith Charts, for example).

So:
Math was invented to handle that which had been discovered.

But:
Any invention is merely the discovery of a way of doing something.

Clarke "invented" the geo-stationary satellite because he "discovered" that the geo-stationary orbit would be a way of doing something that used/followed/obeyed existing physical laws.

And equally:
Some people discover God, but mayhap because someone else invented Her! (And there be only one way to find out! )

Tone

[Bertie]

May 15, 2008 21:08:35 GMT Tone said:

And equally:
Some people discover God, but mayhap because someone else invented Her! (And there be only one way to find out! )

Tone

This always strikes me as a false use of discover.
What is being described is the realisation of a belief in God. Even those, such as Pau,l who undergo some sort of vision, have not discovered something tangible; nor for that matter have they necessarily invented it.
One can invent the wheel or discover Everest, the difference being that the latter was there (placed by God?) whilst it requred a leap of imagination to create the former. Ergo, the wheel could not be discovered; nor Everest be invented.

[Dave M]

In Paul's "mathematics" question, I think we have a definition problem: what do we mean by mathematics? In a maths lesson, we might learn about graphs, and go on to learn how to draw the curve for y=x². Which bits of that are mathematics? The size of the thing which constitutes x² is an existing reality (so there is something to be graphed), but the code we use when we refer to it as "squared", the algebraic use of "x", and the tool which is a graph to display the reality are all inventions.

Reflecting that as a language equivalent, can we perhaps say: "cat" is a noun. Cats exist, but the code "cat" which we use when we refer to them and the tool "noun" which we use to show the possible constructions are both inventions?

[Paul Doherty]

I think Dave has expressed the closest to what I was arguing last night. I like Tone's argument too.

Trouble is, my main domestic opponent (who unfortunately has a Masters in Difficult Maths) argues that even y=x² has an objective truth (or probably she'd prefer 2+2=4, rather than a function), and it would still be true (and would therefore in some form exist) even if man hadn't realised it yet.

She says there is plenty of Maths still waiting to be discovered -- and it really is there, because some of it follows from existing Maths -- we just haven't got there yet. It's definitely there, though, so it must already exist.

We were also arguing about stones and parabolas -- even if we don't know the formula for a parabola, the stone still follows it when thrown -- the parabola doesn't need us to invent it.

I think we're creeping towards god here, aren't we? Blind watchmakers and all. Where's a Dawkins when you need one?

Sue, is Maths a science?

[Barry]

I favour Tone's elegant squaring of the circle .

The Chopin thing is interesting, in that what we mean by 'music' isn't a fixed thing anyway - it's culturally variable. Japanese gagaku (formal court music) often makes its point as much from an instrument being missing as being present; Indonesian gamelan music (to use one example) gets its force not from exact pitch, but from obvious differences in instrument size (and thus pitch and timbre), and their consequent difference in frequency of being sounded.

Neurologists (and anthropologists) have studied how we deal with sound, and the general agreement is that humans are hard-wired (i.e. there is a genetic disposition) to appreciate patterns in sound (changes/recurrences in pitch, timbre, rhythm etc.), but how these patterns are codified as 'music' is culturally-specific.

So, 'music' exists in potentia in any sounds that are made (and therefore could be said to be 'discoverable'); but the discovery is made at a cultural (and organic/subconscious) level - through the proces of a culture's development. Once the culture has a set of phonic 'baselines', then subsequent 'music' is invented/created. Obviously, untangling these two processes (at the dawn of human civilizations) would be a difficult thing to do.

The process of musical composition, in any event, while depending on imagination (which may cary a hint of 'discovery' about it), nonetheless takes place within a relatively strictly defined framework.

[Paul Doherty]

May 16, 2008 0:08:39 GMT Barry said:

imagination (which may cary a hint of 'discovery' about it)

Is that a typo? Surely imagination implies invention?

[Barry]

Nope, I think I meant what I wrote (it was late, guv). I see invention as being the mostly logical process (using existing materials/ideas and constructing something new), although, admittedly, imagination does play a part; but discovery is surely the 'bolt from the blue', where the mind (and imagination) allow us to make a connection that has not previously been considered. The classic (although slightly disproven) leap of imagination that allowed the discovery (not invention) of the structure of DNA, or the 'snake with its tail in its mouth' story of benzene, and the leap of faith that allowed the discovery of 'the Indies' by sailing west, were what sprang to mind.

[Pete]

May 15, 2008 22:21:50 GMT Dave M said:

Reflecting that as a language equivalent, can we perhaps say: "cat" is a noun. Cats exist, but the code "cat" which we use when we refer to them and the tool "noun" which we use to show the possible constructions are both inventions?

Isn't this pretty much what Plato believed? In essence, he said that there was a link between an actualy object and our perception / naming / memory of it. So when you see a cat, then look away, there is still a part of your brain that has now developed some cat-ness, which is how or why you are able to recognise the cat or, indeed, other cats, afterwards. Plato expressed it rather better, I suspect, but not in English.