Relationship between the abstract world of mathematics and the material universe.

[Roehre]

What do you mean by "work out"?

Let x be a square root of 2. x is one (or both) of the two solutions of x^2=2.

That's all there is to it!

But it really becomes interesting as soon as one works out the square root of -1
At that very moment we are entering the world of imaginary numbers.
Literally an imaginary world, but certainly a most important one.
(For the afficionados, that is)

[MrGongGong]

I'm no mathematician
but lets not conflate (that phrase again ..........) Mathematics with Arithmetic

[Serial_Apologist]

but lets not conflate (that phrase again ..........)

Syllogize?

[Boilk]

What is the relationship between the abstract world of mathematics and the material universe?

Does the "abstract world of mathematics" actually exist? Is it not simply man's codification of consistencies he observes in the material universe in order to make some sense of it, and then harnessing those consistencies to his own advantage?

[Sydney Grew]

Several assumptions there:

1) The assumption that a distinction may be drawn, even, between "actual existence" and a "simple codification." The word "simply" there is a kind of value-judgement, designed to belittle the "abstract world of mathematics" in some way.

2) The assumption of a "material universe." As I said recently in another thread, it is sometimes profitable to think oneself back into a pre-Socratic world, with its universe of myths rather than a universe of matter. There is a lamentable tendency in to-day's world, especially among the shallower Anglo-Saxon types, to talk about - indeed to worship - "matter" and to exclude and deprecate both "mind" and "form."

3) So in response to the question "Does the "abstract world of mathematics" actually exist?" the thoughtful man might respond "Does anything actually exist?

[vinteuil]

mathematicians and philosophers of mathematics themselves seem to be divided over this central question - whether mathematics "exists" outside perceived reality, or whether it is merely a reflection of what we can discover "in" that reality. To the extent that mathematicians discover areas of "truth" which are self-consistent - but then which are found to have implications outside the area which the original 'mathematical explorers' were interested in - and then, later on, in apparently totally unrelated fields, are found to be "true" and central to whole new areas of mathematics then under exploration - ie 'reflect a reality' which was not at all in the minds of the original explorers - I am tempted to believe that, yes, mathematics does have a true existence outside our contingent reality.

[Quarky]

I'm no mathematician
but lets not conflate (that phrase again ..........) Mathematics with Arithmetic

Guess this type of discussion tends to be very uneven, since posters will have greatly varying degrees of familiarity with the subject - after all this is a board aimed primarily at Classical Music!

BUT

1. The type of Maths I like is the equation U=0, a single equation to describe all natural phenomena, and where U represents "unworldliness", as described by Richard Feynman in his Book Lectures on Physics - page 25-10, 11 - my bible on physical matters http://student.fizika.org/~jsisko/Kn...lativistic.pdf
Unfortunately, he didn't give a simple solution to the equation.

2. So far as I understand matters, scientists like to make pictures of what is going on, and use maths to prove their point. However there are often situations where more progress is made just by following the logic of the maths, and then trying to make human sense of the results. For example Maxwell when he put forward and solved the equations of electromagnetism, had a very complicated picture of a jelly-type substance supporting electric and magnetic fields (see his 1861 paper). However it subsequently turned out that the equations were correct, and the picture was wrong. So it may be understandable to elevate mathematics to some super-natural standing.

3. What stops me assigning some absolute truth to maths, is that a new explanation of an aspect of nature, frequently involves a devising of a new type of mathematics - e.g theory of fractals to describe natural growth in the material world.

4. I can understand posters tending to denigrate mathematics, since there may be a danger of intellectual arrogance of the true mathematician raising maths to a god-like status - cf. John Cage's views on silence, which I find somewhat arrogant.

[umslopogaas]

As a former scientist, now retired, and one whose weakness at maths was a sore professional trial, with some trepidation I’ll none the less try and comment. I view science as a system of knowledge which operates at two levels.

The first and more immediate is descriptive. Science describes what is observable.

The second and deeper level is explanatory. Science tries to explain what it has described.

It is a fact that when you dig beneath the descriptive surface and try to explain why things are the way they are, if your efforts are successful you will uncover relationships and processes. These in turn can lead to theories. Theories can be tested and if the tests succeed, you end up with laws that have predictive value. At this point, mathematics comes to the fore.

For example, the arrangement of sunflower seeds in a seed head can be described verbally, but the relationship of the curving radial lines has an underlying relationship that can be described mathematically using the Fibonacci Series, which underlies other observable patterns in the natural world and might lead to predictions of the same underlying relationship in other observable phenomena (that’s as far as I go on that one).

For another example, an attempt to pass an electric current through a conducting substance will encounter resistance, large or small depending on whether it is a poor or good conductor. Ohm’s Law relates the factors that determine resistance and allow it to be quantified mathematically for any given situation (I think, its nearly fifty years since my physics O level).

So, as I see it, mathematics allows quantitative measurements of the material world to be used to create laws which can in turn be used to make predictions and monitor and control processes. That is very important in the practical world we inhabit. The maths underlying our understanding and manipulation of electricity ensures that the power company keeps the lights on.

I should say that my own area was crop protection, not physics, and my lack of maths didn’t cripple me, but I wish I’d been better at it, my own observation is that even among fellow applied biologists, the ones with mathematical flair were simple better at the job. Fortunately the main area where I needed maths was in statistical data analysis, and it is so widely recognised that people like me cant do the stats that there is a whole support profession of biometricians on hand to help out. I’ve made extensive use of them!

[MrGongGong]

- cf. John Cage's views on silence, which I find somewhat arrogant.

You can accuse Cage of many things but arrogance is hardly one of them

One of my closest friends has a Phd in algorithmic composition but I wouldn't get him to split the bill after a curry !!

[eighthobstruction]

Jeez....first there was that Zebedee noise "Boing",,,,and then I disappeared....

....What the hell is happening ??....

[Dave2002]

But it really becomes interesting as soon as one works out the square root of -1
At that very moment we are entering the world of imaginary numbers.
Literally an imaginary world, but certainly a most important one.
(For the afficionados, that is)

Refer to msg 14.

Originally Posted by Budapest (modified slightly)
Mathematics is a flawed language (go ask a mathematician to work out the square root of -1).

What do you mean by "work out"?

Let x be a square root of -1. x is one (or both) of the two solutions of x^2=-1.

That's all there is to it!

[Budapest]

Refer to msg 14.

Originally Posted by Budapest (modified slightly)
Mathematics is a flawed language (go ask a mathematician to work out the square root of -1).

What do you mean by "work out"?

Let x be a square root of -1. x is one (or both) of the two solutions of x^2=-1.

That's all there is to it!

Hi Dave, what I was perhaps not very successfully trying to express is the concept of irrational numbers; ie numbers that can not be properly resolved. The square root of two (or indeed 3) is a good example, as is Pi. Using super computers, Pi has been worked-out to trillions of decimal places (3.141592654... ad infinitum) and it's still nowhere near being resolved; ie, the sum does not complete and end. Now, folks might think how does this effect me..? The problem is that all human science is deterministic (including the Turing machines we are all using to take part in this forum), for our science to hold true we need neatly ended sums, whereas nature appears to operate in a different way. As I think I said earlier, mathematicians use all kinds of ingenious methods to get round this, in order to make science work, on paper (higher maths is really just a branch of philosophy). You most definitely do not have to be a mathematician to understand the principles here. I come back to the concept of 'infinity' again (the star filled sky). How do you explain that scientifically? By its nature it's a sum that does not resolve.

Roehre, umslopogaas, Oddball and everyone else who's responded to this thread, all very interesting posts and much food for thought. What does Warwick think of it all? When someone asks about the "Relationship between the abstract world of mathematics and the material universe" I start getting a bit worried (if you want to go insane start thinking about infinity). Are you ok, Warwick?!

[MrGongGong]

As I said i'm not a Mathematician
but from talking with some I find the whole area fascinating once we get away from numbers being about counting objects !
(in the same way that music often becomes more interesting when its freed from narrative )

so (from my primitive understanding !!!) the square root of -1 is i (or -i) which seems fine by me !

[Budapest]

ps. I've just remembered a Melvyn Bragg BBC in Our Time programme about this, and it's still available to listen to:

Melvyn Bragg and guests discuss the nature and existence of mathematical infinity.

http://www.bbc.co.uk/programmes/p0054927

This IOT programme does get a bit technical yet it's worth a listen if you're interested in this stuff. Just remember a concept we can all kind of understand called 'inifinity'. Melvyn Bragg is talking to a bunch of mathematicians who are trying to explain to you, the listener, how they explain the unexplainable. But enough of my explanations, because infinity can not be explained.

[Dave2002]

As I said i'm not a Mathematician
but from talking with some I find the whole area fascinating once we get away from numbers being about counting objects !
(in the same way that music often becomes more interesting when its freed from narrative )

so (from my primitive understanding !!!) the square root of -1 is i (or -i) which seems fine by me !

Mr GG

It gets weirder as it can also be shown that -1 = e ^( i pi). Note that both e and pi are both irrational and transcendental.

http://www.google.com/search?q=e%20is%20transcendental