Archive for the ‘Mathematics’ Category

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The mathematics of jellyfish leaves much to be desired

Published by peter on 2009-06-24 in Decision theory and Mathematics. 0 Comments

A reader of Normblog presents a (standard) constructive argument for the counting numbers and then the infinite cardinals:

I happen to be friends with a jellyfish, called Jelly von Neumann. I asked Jelly about what Professor Atiyah said and she replied as follows…
‘Even if one has never seen any fish, crabs or the like, one may proceed as follows. First consider the empty set, { }, the set which has no elements whatsoever. Call that 0. Next, having got 0, consider the set {0}, whose only element is 0. Call that 1. Next consider the set {0, 1}, whose elements are exactly 0 and 1. Call that 2. Next consider the set {0, 1, 2}. Call that 3.
‘And so on. This gives you the infinite sequence 0, 1, 2, 3,… (One can prove that this sequence is infinite, since the operation involved is injective and never maps anything to 0.) You may even consider the whole infinite set, {0, 1, 2, 3,…}. Call this set omega. And you can go further. For consider the set {omega}. Call this omega + 1. Then consider {omega, omega + 1}, and call this omega + 2. Keep going. You get to omega + omega, and then omega + omega + omega. And so on. Eventually omega squared. Then omega cubed. And so on. Then omega to the power omega. And then (omega to the power omega) to the power omega. And then keep going. Eventually, you get to epsilon-zero. It gets a bit complicated after that. The point is that you can do mathematics just by virtue of thinking. Of course, I am a rather special jellyfish in that regard.’

Let us look carefully at the first few lines.  Before we have defined or constructed a single number, we are expected to have available a notion of a set and a notion of an element of a set.

First consider the empty set, { }, the set which has no elements whatsoever.

This is very odd – we are people who apparently know some set theory, but we cannot yet count (since we have not yet constructed the counting numbers).   And not just any set, but a set with no elements.   So maybe we can count!  How else can we tell that there are no elements in the empty set?  Perhaps we can only count zero objects.   And, moreover, this set is called “the empty set”, so presumably we know that there is only one of them.  There’s some pretty advanced set theory right there, in that casual statement of uniqueness, I would say.  (The claim of uniqueness, however, is not required for Jelly’s construction.)
Putting aside the question whether it is possible in principle for anyone, even those us with access to counting numbers, to count zero objects (arguably, counting is by definition an activity which requires the presence of at least one object to occur), let us continue with Jelly’s argument:

Call that 0.

So we can label objects.

Next, having got 0,

Wait a goddam minute, buster!  We just labeled an object “{  }” with the label “0“.    That is something different from getting or having anything.   And surely, in order to label an object “{  }” with a label “0“, we must in some fundamental sense already had had the label  “0“.   If we did not already have it, how else could we use it to label an object?   Jelly is using some pretty sleazy slight-of-hand here to slip from assigning a label that looks like a counting number to having the counting number itself, ready and able to be used for counting.   If the label we had used was (say) the greek letter alpha, then Jelly’s argument would proceed in exactly the same way as before, but we would not end the argument having defined the counting numbers.
Ignoring these problems, let us proceed:

consider the set {0}, whose only element is 0.

So now “0″ is an object, available for use as the element of a set. And we not only know some set theory, we ALSO know how to construct sets!   Just how do we do this?  Do we pick the object (or the label?) called “0″ and put it inside some curly braces?  How do we know when to start and stop picking objects?  For some reason we picked just one object.  Do we know how to count already?  At the next step we construct a set with two objects:

Next consider the set {0, 1}, whose elements are exactly 0 and 1. Call that 2.

From what collection of objects (or labels?) did we select the one called “0″ , or (respectively) the ones called “0″ and “1″? We seem not only able to construct sets and to count objects, but we also know how to select particular objects (not just any old objects, but particular ones) from some undefined collection of objects. Quite some skills we have here, we people who don’t yet know how to count.  And is the object that is here called “0” a different object with the same label as the one called “0” just three sentences before?   If they are different, how many of these different objects with the same label do we have?  And how can we tell them apart?  And, if they are not different, we must be re-using the same object called “0”.  Can we do this?  When last handled by us (two sentences before), the object called “0” was sitting inside the set {0}.  Can we just up and take it out and plonk it down inside the set {0,1}?  There are lots of deep questions here, questions whose possibly-different answers motivate entire branches of pure mathematics (e.g., linear logic, which deals with formal logics where we have available only a fixed and finite number of each mathematical symbol), which our jellyfish-cum-mathematician is glossing over or ignoring.
After a few rounds of this, Jelly hits us with:

And so on. This gives you the infinite sequence 0, 1, 2, 3,…

Well, no, actually. We never get an infinite sequence, since we, in this universe, can only ever complete a finite number of such steps in our lifetimes.  This is true even if all humans who ever lived, who are living, and who ever will live were to add their tuppence-worth of steps to the argument.  It’s hard to have confidence in a jellyfish claiming to construct a collection of infinite cardinals who can’t seem to distinguish between a finite and an infinite sequence.  At best (modulo the flaws identified above) we could get a finite, ever-growing sequence of counting numbers, a sequence that can be proven to exceed any pre-determined numerical threshold (thinking of these labels as real numbers for the moment), provided we allow sufficient time for the steps to be undertaken in the order described.  A finite, ever-growing sequence is not ever an infinite sequence; at best, we might call it potentially-infinite.
I think Mr Jelly ought to forget the peano lessons and adopt a cat.   And Norm, a Zimbabwean by birth, could perhaps remember how difficult it is to count objects in chiShona, with its ostentatious plenitude of noun-classes (21 according to Dale), and associated multitudes of counting words; urban Shona children nowadays usually count in English, even when they know little other English.
Reference:
D. Dale [1968]:  Shona Companion. Mambo Press, Gweru, Zimbabwe.  Second edition, 1972.

Achilles and the Tortoise

Published by peter on 2009-06-07 in Argumentation, Mathematics and Matherati. 0 Comments

An amusing account (at least to a mathematician) by Harvey Friedman of an encounter with eccentric Russian mathematician and dissident Alexander Yessenin-Volpin. Friedman supervised the Stanford PhD of John E. Hutchinson, who taught me calculus.  (Hat tip: AB)

Let me give an example. I have seen some ultrafinitists go so far as to challenge the existence of 2^100 as a natural number, in the sense of there being a series of ‘points’ of that length. There is the obvious ‘draw the line’ objection, asking where in , . . . , 2^100 do we stop having ‘Platonistic reality’? Here this . . . is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist [mathematician Alexander] Yessenin Volpin during a lecture of his.  He asked me to be more specific.  I then proceeded to start with 2^1 and asked him whether this is ‘real’ or something to that effect.  He virtually immediately said yes.  Then I asked about 2^2, and he again said yes, but with perceptible delay. Then 2^3, and yes, but with more delay.  This continued for a couple of more times, till it was obvious how he was handling this objection.  Sure, he was prepared to always answer yes, but he was going to take 2^100 times as long to answer yes to 2^100 than he would to answering 2^1.  There is no way that I could get very far with this. (pp. 4-5).

Note: Of course, Friedman is wrong about the . . . being replaced by 100 items. We would expect it to be replaced with just 96 items, since 4 items in the list of 100 are already listed explicitly.

Reference:

Harvey M. Friedman [2002]: Lecture Notes on Philosophical Problems in Logic. Princeton University.

Complexity of communications

Published by peter on 2008-08-17 in Advertising, Anthropology, Art, Mathematics and Telecommunications. 0 Comments

Recently, I posted about probability theory, and mentioned its modern founder, Andrei Kolmogorov.  In addition to formalizing probability theory,  Kolmogorov also defined an influential approach to assessing the complexity of something.
He reasoned that a more complex object should be harder to create or to re-create than a simpler object, and so you could “measure” the degree of complexity of an object by looking at the simplest computer program needed to generate it.  Thus, in the most famous example used by complexity scientists, the 1915 painting called “Black Square” of Kazimir Malevich, is allegedly very simple, since we could recreate it with a very simply computer program:
Paint the colour black on every pixel until the surface is covered, say.

But Kolmogorov’s approach ignores entirely the context of the actions needed to create the object.   Just because an action is simple or easily described, does not make it easy to do, or even easy to decide to do.   Art objects, like most human artefacts, are created with deliberate intent by specific creators, as anthropologist Alfred Gell argued in his theory of art.  To understand a work of art (or indeed any human artefact) we need to assess its effects on the audience in the light of its creator’s intended effects, which means we need to consider the intentions, explicit or implicit, of its creators.  To understand these intentions in turn requires us to consider the context of its creation, what a philosopher of language might call its felicity conditions.
Malevich’s Black Sqare can’t be understood, in any sense, without understanding why no artist before him created such a painting.  There is no physical or technical reason that Rembrandt, say, or Turner, could not have painted a canvas consisting only of one colour, black.  But they did not, and could not have, and could not even have imagined doing so. (Perhaps only the 18th-century Welsh painter Thomas Jones could have imagined doing so, with his subtle paintings of near-monochrome Neapolitan walls.)  It is not a coincidence that Malevich’s painting appeared in the historical moment when it did, and not anytime before nor anyplace else.   For instance, Malevich worked at a time when educated people were fascinated with notions of a fourth or even further dimensions, and Malevich himself actively tried to represent these other dimensions in his art.  To imagine that such a painting could be adequately described without reference to any art-historical background, or socio-political context, or the history of ideas is to confuse the syntax of the painting with its semantics and pragmatics.  We understand nothing about the painting if all we understand is that every pixel is colored black.
We have been here before.  The mathematical theory of communications of Claude Shannon and Warren Weaver has been very influential in the design of the physical layers of telecommunications and computer communications networks.   But this theory explicitly ignores the semantics – the meanings – of messages. (To be fair to Shannon and Weaver they do tell us explicitly early on that they will be ignoring the semantics of messages.)    Their theory is therefore of no use to anyone interested in communications at layers above the physical transmission of signals, that is, anyone interested in understanding or using communication to communicate with other people or machines.
References:
M. Dabrowski [1992]: “Malevich and Mondrian:  nonobjective form as the expression of the “absolute”. ” pp. 145-168, in: G. H. Roman and V. H. Marquardt (Editors): The Avant-Garde Frontier:  Russia Meets the West, 1910-1930. Gainesville, FL, USA: University Press of Florida.
Alfred Gell [1998]: Art and Agency:  An Anthropological Theory.  Oxford, UK: Clarendon Press.
L. D. Henderson [1983]: The Fourth Dimension and Non-Euclidean Geometry in Modern Art. Princeton, NJ, USA: Princeton University Press.
Claude E. Shannon and Warren Weaver [1963]: The Mathematical Theory of Communication. Chicago, IL, USA:  University of Illinois Press.