The Uses of Infinity—Emergenceand Reduction Reconciled?

Is the whole greater than the sum of theparts? Holism or reduction?

1. The jargonEmergence as behaviour that is both novel androbust, especially in comparison to a theoryof the microscopic details. (So emergence isnot just ‘good’ variables and-or approximationschemes.)

Reduction: The core idea is that one theoryis a definitional extension of another.

Theory Tb = Tbottom/basic/best reducesTt = Ttop/tangible/tainted; or Tt reducesto, is reducible to, Tb

when for all words used by Tt, there are defin-itions in the language of Tb such that when allthe definitions are added to Tb, you can deriveall of Tt. (The definitions may be very hard tofind; and very long.)

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Example?: Thermodynamics = Tt uses ‘tem-perature’, ‘pressure’ etc.; statistical mechanics= Tb uses ‘position’, ‘momentum’, ‘molecule’,‘probability’.

Supervenience: Total matching of any twoobjects as regards one family of properties (calledthe subvening family, say B) implies their to-tal matching as regards the other family (thesupervening family, say T ).

Example?: The mental supervenes on thephysical: if a cat sees yellow, an atom-for-atomreplica of the cat also sees yellow.

Supervenience is a weakening of definitionalextension: it allows one or more of the defin-itions (of a property T ∈ T in terms of thevarious B ∈ B) to be infinitely long: an infi-nite disjunction (...or...or...or......) of “ways tobe T”.

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2. Two claims(1) Emergence is compatible with definitionalextension.I will give three examples, each with a para-meter N = ∞; (N is the number of ‘degreesof freedom’).And for each: choosing a salient weaker theoryusing finite N blocks the reduction.

(2) Supervenience is a red herring. It is sci-entifically useless because it gives no control onthe infinite disjunction; in particular, no kindof limit is taken.

Compare continuous models of fluids (e.g.sound, flow), which model a finite system, i.e asystem with finitely many degrees of freedom(atomism!), as infinite.

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Three obvious justifications of N = ∞, whichare shared with such models of fluids.

1: Mathematical convenience: impossible tooverstate!

2: Elimination of finitary effects.3: Empirical success: the proof of the pud-

ding is in the eating.

So, agreed: ∞ − 1023 = ∞ and N is actu-ally finite! But if f (1023) ≈ f (1046) etc, thenit is good, and sometimes even indispensible,to model the system with f (∞).

Agreed: maybe reduction needs more thanjust definitional extension; maybe the definedproperties have to be already in, or even cen-tral to, the reducing theory. But in our exam-ples, Tb rich enough to consider N = ∞ doescontain the defined properties.

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Figure 1: The method of arbitrary functions

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3. The method of arbitrary functionsA roulette wheel, with unknown biassing inspin and friction. But suppose we assume:

(i): there are a large number N of alternat-ing arcs of red and black;

(ii): the biassing favours and disfavours bigsegments;

(iii): within a single big segment, the bias is“smooth”: adjacent arcs get a similar bias.

Then we can be confident that each long-run frequency is about 50%. For any biassingregime, no matter how wiggly (sensitive to an-gular position), can be washed out so as to giveequiprobability, by considering a sufficientlylarge N .

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Indeed:

For any M ∈ IR, for all probability den-sity functions f with derivative boundedby M , | f ′ |< M : as N = the numberof arcs, goes to infinity:∫R f dµ ≡ prob(Red) → 1

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Emergent probabilitiesThe equiprobability is robust, i.e. holds formany different density functions. And Tt isa definitional extension of Tb, if we take Tb tobe a rich enough model of the wheel to includeboth

(i) the postulation of various possible densityfunctions f ; and

(ii) consideration of the infinite limit N =∞.

The emergent behaviour, i.e. equiprobability,is frustrated if we confine ourselves to finitaryTb.

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4. FractalsFractals form a recent episode in a grand nar-rative across 400 years of mathematics, thegeneralization of the idea of a mathematicalfunction: Galileo, Euler—and Coverly.

Philosophy is written in this immense bookthat stands ever open before our eyes (I speakof the Universe), but it cannot be read if onedoes not first learn the language and recog-nize the characters in which it is written. Itis written in mathematical language, and thecharacters are triangles, circles and other geo-metrical figures, without the means of which itis humanly impossible to understand a word;without these philosophy is a confused wander-ing in a dark labyrinth. (Galileo, Il Saggiatore1623, Opere VI, 197.)

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d’Alembert vs. Euler, on vibrating strings:d’Alembert (1747) describes the displacementf (x, t) of a vibrating string by:

∂2f

∂t2= a2 ∂2f

∂x2.

Can this describe a plucked string, with a “cor-ner”? d’Alembert says: No.

Euler replies (1748): Yes. The analysis shouldbe generalised ‘so that the initial shape of thestring can be set arbitrarily ... either regularand contained in a certain equation, or irreg-ular and mechanical’.

Truesdell: ‘Euler’s refutation of Leibniz [i.e.:Leibniz’s claim that natural phenomena canall be described by what we now call analyticfunctions] was the greatest advance in scien-tific methodology in the entire century.’

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Thomasina: ... Each week I plot your equa-tions dot for dot, and they draw themselvesas commonplace geometry, as if the world offorms were nothing but arcs and angles. God’struth, Septimus, if there is an equation for abell, then there must be an equation for a blue-bell: and why not a rose? ... Do we believenature is written in numbers?

Septimus: We do.

Thomasina: Then why do your equationsonly describe the shapes of manufacture? ...Armed thus, God could only make a cabinet.

Septimus: He has mastery of equations whichlead into infinities where we cannot follow.

Thomasina: What a faint heart! We mustwork outward from the middle of the maze.We will start with something simple. I willplot this leaf and deduce its equation. Youwill be famous for being my tutor when LordByron is dead and forgotten.

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We focus on the idea that a set of spatialpoints can have a dimension that is not aninteger. For us: scaling dimension.

Figure 2: Dimension as an exponent

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A square with edge l is the union of l2 unitsquares. For example, a square whose edge isl = 2 units long is the union of 22 = 4 unitsquares.

A cube with edge l is the union of l3 unitsquares: a cube whose edge is l = 2 units longis the union of 23 = 8 unit cubes. Indeed:

no. units in object with edge l = ldim of object.

Applying this to more “pathological” objects,we find that they have non-integer dimensions.

Objects like the Cantor set (1872) are self-similar: they are unions of shrunken copiesof themselves—just as much as a square or acube is...

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Figure 3: The Cantor set

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So what power of 3 is equal to 2? At thispoint, we recall some elementary logarithms!The idea is that 102 = 100 is re-written aslog10 100 = 2. This implies that 2 = 3(log 2/ log 3).So the dimension of C is log 2/ log 3: which isabout 0.63.

Similarly for the Koch snowflake. Each “side”is the union of four smaller similar curves, eachsmaller by a factor 3. So applying again theidea of dimension as an exponent, we get:

4 = 3dimension of Koch.

So we get:

dimension of Koch =log 4

log 3≈ 1.26.

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Figure 4: The Koch snowflake

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Emergent dimensionsNon-integer dimensions are novel. And theyare ‘robust’ in various senses. So indeed: wehave emergent dimensions.

Take as Tb: the rich modern theory of scalingdimension (and its cousin notions); and as Tt:the assignment of non-integral dimensions toobjects like the Cantor set.

Clearly, Tb contains Tt! But if Tb is just the(salient) traditional theory of dimension, thereis no reduction.

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Is Fractal Geometry the Geometry of Na-ture?Distinguish two questions. First: do fractalsdescribe the geometry of naturally occurring(“natural history”) objects?

Agreed: It looks like it! Thus in Arcadia:

Valentine: If you knew the algorithm andfed it back say ten thousand times, each timethere’d be a dot somewhere on the screen. You’dnever know where to expect the next dot. Butgradually you’d start to see this shape, becauseevery dot would be inside the shape of this leaf.It wouldn’t be a leaf, it would be a mathemat-ical object. But yes ...

Figure 5: A fern growing dot by dot

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But a leaf and its ilk have no tower of struc-ture, on ever-smaller length-scales. So: ‘No’.

Agreed: (i) Often a property obeys a powerlaw, with a resolution raised to a non-integerpower; i.e. the exponent is not an integer—asoccurs for dimension, with fractals.

(ii): It is heuristically better to have the sug-gestive language and results of fractal geome-try, than just the power law.

Indeed, ever since Lagrange introduced con-figuration space (1788), physical theories havemade indispensable use of various spaces equippedwith structures that surely deserve the name‘geometry’. So to the second question—Dosome of our best physical theories use frac-tals to describe certain subsets of their ab-stract spaces?—the answer is: Yes.

Example: Statistical mechanics describes as-pects of some processes with scale-free (regimesof) theories, involving power-law behaviour onall scales, and fractals.

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5. Phase transitionsStatistical mechanics follows thermodynamicsin representing phase transitions (like boilingand freezing) by non-analyticities, “disconti-nuities” in a function (the free energy F ).

But these cannot occur for a finite system.So the thermodynamic limit is taken: N :=number of particles, V := volume → ∞ withρ = N/V fixed.

This infinite limit brings new mathematicalstructure. Again, we have three obvious jus-tification: mathematical convenience, elimina-tion of finitary effects, and empirical success.

But what exactly should we say about our(finite!) kettle?

I endorse Mainwood’s proposal: for systemswith a well-defined thermodynamic limit, FN →F∞, we say: phase transitions occur in the fi-nite system iff F∞ has non-analyticities.

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And if we wish, we can add: and if N is largeenough, or the gradient of FN is steep enough.This vagueness is acceptable.

So the emergent phase transitions are re-ducible: to a sufficiently rich theory Tb thattakes the appropriate infinite limit; but not toa theory using finite N .

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6. Other examplesThere are many other examples of novel androbust behaviour in a limit. Some are relatedto our examples: e.g. superselection in theN →∞ limit of quantum mechanics.

Others are based on the idea of a continuousmodel of a fluid. Yet others involve examin-ing wave phenomena in the limit of very shortwavelengths:

1: the geometric optics limit (λ → 0) of waveoptics; and similarly

2: the classical limit (~ → 0) of quantummechanics; (or rather: some aspects of thislimit! Which involves so much else, such as:coherent states, decoherence, the measurementproblem ...)

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