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21 December 2011

Do numbers "exist"?

I've previously suggested that mathematics might be the underlying
foundation of reality, and this has (perhaps not unreasonably) been
viewed by some as a rather provocative thing to say. I'll edit this
post later to provide some links, but in the meantime it might be
helpful to think about poor old Pi. As every schoolchild knows, the
value of Pi is 3.14159etc and cannot be accurately represented as a
ratio between two integers; there will always be a remainder, however
small. Some people argue that Pi is merely a consequence of the
Euclidean geometry we use to describe space, and therefore merely a
consequence of rules we set up in advance.

I'm not so sure - I think there is something fundamentally spooky
about numbers like Pi, which crops up all over the place in areas
where geometry is neither here nor there. It occurs in probability
theory, complex number calculations (which I suppose can be considered
as Euclidean, but hey), quantum mechanics, thermodynamics - pretty
much any area of mathematics and physics, regardless of any axiomatic
specification of the nature of the space into which we draw circles.
Where I think part of the problem lies is that people commonly think
that Pi is *defined* by the relationship between a circumference of a
circle and its diameter, whereas I would simply say that we can
calculate the value to be equivalent to Pi, but Pi itself (and 2*Pi,
or "Tau", which some argue is a more relevant quantity) has a sort of
independent "existiness" that does not depend on Euclid's axioms, for
example. Pi is a property of integer mathematics, and it is hard to
get much more basic than that.

Philosophically (aarrgghh!) this is a tricky one. Some people, such as
the renowned "Slicer" at the blog http://t-rinder.typepad.com, suggest
that mathematics and logic need an underpinning deity of sorts (who of
course pitches out as the JudeoChristian construct commonly known as
"God", but that's a WTF for another day). Maybe I am blind or too
dismissive or hard-hearted, and I cannot appreciate the very clever
subtlety of this suggestion, but it seems like crap to me. Even
Stephen Hawking approvingly quotes the old canard "God created the
integers" in his classically wry fashion, so I'm not sure whether
Hawking is agreeing or disagreeing with my feeling (let's put it no
more strongly than that) that numbers have a kind of existiness that
is different from the existiness of chairs, tables, donkeys and
bloggers.

Anyway, this is to throw things open to the crowd out there
(especially @bagguley - comments most welcome!) to see if anyone can
help clarify my simple brain, because Slicer (no, that's not his real
name - it may be taken from something he got a head injury from in
childhood) thinks that I am epistemiologically rudderless and only
capable of insulting his wee heroes. Have at it, people (if indeed
there is anyone there...)!

10 comments:

  1. Again - to correct misrepresentation - as anyone who reads my replies to shanemuk's assertions will see (comments 7 & 12, of 21), I have NOT argued that either maths or logic NEEDS an underlying deity; merely that shanemuk's claim that "maths is all" and needs no explanation for its presence/usefulness, is not the view of some mathematicians who, I respectfully suggest know a little more maths than shanemuk. I have also stated that I accept that such a notion, held by both theist and non-theist mathematicians, falls well short of an evidence base for deus or theos, never mind a Christian concept of God.
    Slicer notes (from Twitter feeds) that use of CAPITALS seems to offend shanemuk (who is happy to use ** for emphasis in his posts), and apologises for any offence cause by their use - none intended - merely a device to highlight points made previously, and key points.

    Flattered to have provoked so many comments from this blogger, tho...

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  2. Great! Slicer exists too :-) I would urge interested readers to go and visit Slicer's thread because it is most interested. Slicer does of course eschew the notion of appealing to the authority of various other mathematicians, at least when challenged in this regard, but he is quite happy to at least make it look like little old Shanemuk is not permitted (by Slicer of course) to challenge Slicer's interpretation of what these venerable punters are saying.

    Another interesting (and slightly philosophically unnerving) tendency for Slicer is to suggest that just because I can't *prove* that his assertion that god is "intimately connected" with maths/logic (how does that sit?), nor can I completely logically *refute* it, that somehow it remains a reasonable position. I really don't see how Slicer makes a distinction between issuing a valid hypothesis relating to the nature of maths and space pixies, and simply pulling some guff out of the aether. It's just not clear at all. Slicer could make any meaningless assertion (and he does quite a lot of this), and somehow I am supposed to simply take this as a valid proposition. Anyway, read the thread if you have the stomach for it, and in the meantime, I invite people with a better grasp of logic and irreverent turn of phrase to offer opinions on whether mathematics/logic is truly universal/basic, or whether it is simply the arbitrary whim of the Grand Celestial Elf.

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  3. (InterestING, of course - apols for the typo)

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  4. I think I'll invoke the anthropic principle here, and suggest that it might only be possible for conscious, self-aware intelligence of our type to emerge in a universe which follows mathematically based laws. Or, maybe, appears to follow mathematically based laws.

    Of course there's a problem here with "standard" multiverse models, as they are also mathematical in nature, and just generate just more mathematical universes. The "multiverse" is now generally defined as the set of all possible universes (albeit there's a wealth of different postulates you can use to generate your multiverse). However, I suspect that it's just the set of all mathematically definable universes (or even a subset of those). However, that might simply be due to limitations of our perceptions. There may be more beyond (but, then, set is a mathematical concept so...).

    nb. if the universe is truly mathematical and works that way, where does that leave Godel and the incompleteness theorem? After all, that allows for some things to be true yet unprovable through mathematical mechanism. If, as you suggest, the universe is literally working mathematics, what about those parts that are true yet the universe can't generate them?

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  5. Part of this comes back to the notion of "Turing completeness" - it can be proven that the Universe can act as a Universal Turing Machine, so is in principle capable of carrying out any computable operation. Goedel incompleteness is therefore a feature of mathematics in general, not the universe specifically, but I don't think this creates a problem for us. Indeed, I would even go a wee bit further and tentatively suggest that *integer* mathematics underpins reality, and as long as we have the very basic axioms of integer maths, that can be reduced to the Boolean principles of binary, we have all we need to "make" a universe.

    Whether Goedel remains a problem, therefore, I don't entirely know. I would point out that my suggestion is not that the universe IS all of mathematics, but rather that the universe is a specific mathematical "object" in the same way as a single instance of the "Game of Life" or the Fibonacci Sequence or the Mandelbrot Set etc, so although Goedel incompleteness is a puzzling johnny and no mistake, I don't think it need worry us in this regard :-)

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  6. Hi Helio. I don't think you can shrug of Godel so easily, or rather your model just shifts the problem down a level. If you say maths is reality, or underpins it, at some point you have to grapple with Incompleteness and the only way out I can see is invoking something else at level zero that is not maths. For me that puts us into the same philosophical territory as the uncaused cause and infinite regress and Godel of the gaps.

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  7. Carrying on from my tweets, I still think that maths depends on the axioms you make up front.

    E.g. does A * B = B * A

    In some types of maths, it doesn't, in others, it does.

    Some axioms produce more useful results than others but numbers in no way "exist" as a special entity.

    3.1415... is a common ratio/number for maths with certain axioms and isn't in others, where a different number would crop up. To claim that, just because it occurs a lot, then it is somehow magically transcendent is a total non-sequitur.

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  8. I take the point, but is it not the case that all forms of maths are remappable, at least in principle, to Boolean integer logic? In that way of looking at things, all we are doing when we apply axioms is simply redefining operators. None of this alters the underlying logic, and it is from that straight boolean logic that Pi inexorably arises (and e and the "golden ratio" and the rest of 'em).

    Now maybe I'm wrong in this, but (help me out here! :-) I still can't get away from the feeling (better expressed by Penrose, in fairness, as in most things) that there is something pretty fundamental about integers and addition, and that these are not arbitrary conventions, but represent something very basic.

    And I do think this relates to Turing completeness, which is a more relevant measure than Goedel completeness, but I so totally would need a proper mathematician (and maybe even a mathematical philosopher!) to tease this all out. Thanks guys for your comments - much appreciated :-)

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  9. A quick google reveals that Tegmark himself gets round the Godel objection by placing a limit on the complexity of the mathematical structures that have physical existence to those that are Godel Complete. So instead of a Mathematical Universe we have a computable one. Even then he admits it faces serious challenges, not least that virtually all successful theories of physics violate it.

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  10. The existence of numbers, integers at least, probably exists because of human biology, because we see patterns and similarities. It's quite possible that other species see every item as an individual item, distinguishable from all other similar items and never confused with another no matter how similar. We see trees and rocks and giraffes and identify the similarities. From this we created the concept of numbers.

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