Triadomania. Thought of the Day 117.0

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Peirce’s famous ‘triadomania’ lets most of his decisive distinctions appear in threes, following the tripartition of his list of categories, the famous triad of First, Second, and Third, or Quality, Reaction, Representation, or Possibility, Actuality, Reality.

Firstness is the mode of being of that which is such as it is, positively and without reference to anything else.

Secondness is the mode of being of that which is such as it is, with respect to a second but regardless of any third.

Thirdness is the mode of being of that which is such as it is, in bringing a second and third into relation to each other.

Firstness constitutes the quality of experience: in order for something to appear at all, it must do so due to a certain constellation of qualitative properties. Peirce often uses sensory qualities as examples, but it is important for the understanding of his thought that the examples may refer to phenomena very far from our standard conception of ‘sensory data’, e.g. forms or the ‘feeling’ of a whole melody or of a whole mathematical proof, not to be taken in a subjective sense but as a concept for the continuity of melody or proof as a whole, apart from the analytical steps and sequences in which it may be, subsequently, subdivided. In short, all sorts of simple and complex Gestalt qualities also qualify as Firstnesses. Firstness tend to form continua of possibilities such as the continua of shape, color, tone, etc. These qualities, however, are, taken in themselves, pure possibilities and must necessarily be incarnated in phenomena in order to appear. Secondness is the phenomenological category of ‘incarnation’ which makes this possible: it is the insistency, then, with which the individuated, actualized, existent phenomenon appears. Thus, Secondness necessarily forms discontinuous breaks in Firstness, allowing for particular qualities to enter into existence. The mind may imagine anything whatever in all sorts of quality combinations, but something appears with an irrefutable insisting power, reacting, actively, yielding resistance. Peirce’s favorite example is the resistance of the closed door – which might be imagined reduced to the quality of resistance feeling and thus degenerate to pure Firstness so that his theory imploded into a Hume-like solipsism – but to Peirce this resistance, surprise, event, this thisness, ‘haecceity’ as he calls it with a Scotist term, remains irreducible in the description of the phenomenon (a Kantian idea, at bottom: existence is no predicate). About Thirdness, Peirce may directly state that continuity represents it perfectly: ‘continuity and generality are two names of the same absence of distinction of individuals’. As against Secondness, Thirdness is general; it mediates between First and Second. The events of Secondness are never completely unique, such an event would be inexperiencable, but relates (3) to other events (2) due to certain features (1) in them; Thirdness is thus what facilitates understanding as well as pragmatic action, due to its continuous generality. With a famous example: if you dream about an apple pie, then the very qualities of that dream (taste, smell, warmth, crustiness, etc.) are pure Firstnesses, while the act of baking is composed of a series of actual Secondnesses. But their coordination is governed by a Thirdness: the recipe, being general, can never specify all properties in the individual apple pie, it has a schematic frame-character and subsumes an indefinite series – a whole continuum – of possible apple pies. Thirdness is thus necessarily general and vague. Of course, the recipe may be more or less precise, but no recipe exists which is able to determine each and every property in the cake, including date, hour, place, which tree the apples stem from, etc. – any recipe is necessarily general. In this case, the recipe (3) mediates between dream (1) and fulfilment (2) – its generality, symbolicity, relationality and future orientation are all characteristic for Thirdness. An important aspect of Peirce’s realism is that continuous generality may be experienced directly in perceptual judgments: ‘Generality, Thirdness, pours in upon us in our very perceptual judgments’.

All these determinations remain purely phenomenological, even if the later semiotic and metaphysical interpretations clearly shine through. In a more general, non-Peircean terminology, his phenomenology can be seen as the description of minimum aspects inherent in any imaginable possible world – for this reason it is imaginability which is the main argument, and this might point in the direction that Peirce could be open to critique for subjectivism, so often aimed at Husserl’s project, in some respects analogous. The concept of consciousness is invoked as the basis of imaginability: phenomenology is the study of invariant properties in any phenomenon appearing for a mind. Peirce’s answer would here be, on the one hand, the research community which according to him defines reality – an argument which structurally corresponds to Husserl’s reference to intersubjectivity as a necessary ingredient in objectivity (an object is a phenomenon which is intersubjectively accessible). Peirce, however, has a further argument here, namely his consequent refusal to delimit his concept of mind exclusively to human subjects (a category the use of which he obviously tries to minimize), mind-like processes may take place in nature without any subject being responsible. Peirce will, for continuity reasons, never accept any hard distinction between subject and object and remains extremely parsimonious in the employment of such terms.

From Peirce’s New Elements of Mathematics (The New Elements of Mathematics Vol. 4),

But just as the qualities, which as they are for themselves, are equally unrelated to one other, each being mere nothing for any other, yet form a continuum in which and because of their situation in which they acquire more or less resemblance and contrast with one another; and then this continuum is amplified in the continuum of possible feelings of quality, so the accidents of reaction, which are waking consciousnesses of pairs of qualities, may be expected to join themselves into a continuum. 

Since, then an accidental reaction is a combination or bringing into special connection of two qualities, and since further it is accidental and antigeneral or discontinuous, such an accidental reaction ought to be regarded as an adventitious singularity of the continuum of possible quality, just as two points of a sheet of paper might come into contact.

But although singularities are discontinuous, they may be continuous to a certain extent. Thus the sheet instead of touching itself in the union of two points may cut itself all along a line. Here there is a continuous line of singularity. In like manner, accidental reactions though they are breaches of generality may come to be generalized to a certain extent.

Secondness is now taken to actualize these quality possibilities based on an idea that any actual event involves a clash of qualities – in the ensuing argumentation Peirce underlines that the qualities involved in actualization need not be restrained to two but may be many, if they may only be ‘dissolved’ into pairs and hence do not break into the domain of Thirdness. This appearance of actuality, hence, has the property of singularities, spontaneously popping up in the space of possibilities and actualizing pairs of points in it. This transition from First to Second is conceived of along Aristotelian lines: as an actualization of a possibility – and this is expressed in the picture of a discontinuous singularity in the quality continuum. The topological fact that singularities must in general be defined with respect to the neighborhood of the manifold in which they appear, now becomes the argument for the fact that Secondness can never be completely discontinuous but still ‘inherits’ a certain small measure of continuity from the continuum of Firstness. Singularities, being discontinuous along certain dimensions, may be continuous in others, which provides the condition of possibility for Thirdness to exist as a tendency for Secondness to conform to a general law or regularity. As is evident, a completely pure Secondness is impossible in this continuous metaphysics – it remains a conceivable but unrealizable limit case, because a completely discon- tinuous event would amount to nothing. Thirdness already lies as a germ in the non-discontinuous aspects of the singularity. The occurrences of Secondness seem to be infinitesimal, then, rather than completely extensionless points.

Causal Isomorphism as a Diffeomorphism. Some further Rumination on Philosophy of Science. Thought of the Day 82.0

Let (M, gab) and (M′, g′ab) be (temporally oriented) relativistic spacetimes that are both future- and past-distinguishing, and let φ : M → M′ be a ≪-causal isomorphism. Then φ is a diffeomorphism and preserves gab up to a conformal factor; i.e. φ⋆(g′ab) is conformally equivalent to gab.

Under the stated assumptions, φ must be a homeomorphism. If a spacetime (M, gab) is not just past and future distinguishing, but strongly causal, then one can explicitly characterize its (manifold) topology in terms of the relation ≪. In this case, a subset O ⊆ M is open iff, ∀ points p in O, ∃ points q and r in O such that q ≪ p ≪ r and I+(q) ∩ I(r) ⊆ O (Hawking and Ellis). So a ≪-causal isomorphism between two strongly causal spacetimes must certainly be a homeomorphism. Then one invokes a result of Hawking, King, and McCarthy that asserts, in effect, that any continuous ≪-causal isomorphism must be smooth and must preserve the metric up to a conformal factor.

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The following example shows that the proposition fails if the initial restriction on causal structure is weakened to past distinguishability or to future distinguishability alone. We give the example in a two-dimensional version to simplify matters. Start with the manifold R2 together with the Lorentzian metric

gab = (d(at)(db)x) − (sinh2t)(dax)(dbx)

where t, x are global projection coordinates on R2. Next, form a vertical cylinder by identifying the point with coordinates (t, x) with the one having coordinates (t, x + 2). Finally, excise two closed half lines – the sets with respective coordinates {(t, x): x = 0 and t ≥ 0} and {(t, x): x = 1 and t ≥ 0}. Figure shows, roughly, what the null cones look like at every point. (The future direction at each point is taken to be the “upward one.”) The exact form of the metric is not important here. All that is important is the indicated qualitative behavior of the null cones. Along the (punctured) circle C where t = 0, the vector fields (∂/∂t)a and (∂/∂x)a both qualify as null. But as one moves upward or downward from there, the cones close. There are no closed timelike (or null) curves in this spacetime. Indeed, it is future distinguishing because of the excisions. But it fails to be past distinguishing because I(p) = I(q) for all points p and q on C. For all points p there, I(p) is the entire region below C. Now let φ be the bijection of the spacetime onto itself that leaves the “lower open half” fixed but reverses the position of the two upper slabs. Though φ is discontinuous along C, it is a ≪-causal isomorphism. This is the case because every point below C has all points in both upper slabs in its ≪-future.

Dialectics of God: Lautman’s Mathematical Ascent to the Absolute. Paper.

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Figure and Translation, visit Fractal Ontology

The first of Lautman’s two theses (On the unity of the mathematical sciences) takes as its starting point a distinction that Hermann Weyl made on group theory and quantum mechanics. Weyl distinguished between ‘classical’ mathematics, which found its highest flowering in the theory of functions of complex variables, and the ‘new’ mathematics represented by (for example) the theory of groups and abstract algebras, set theory and topology. For Lautman, the ‘classical’ mathematics of Weyl’s distinction is essentially analysis, that is, the mathematics that depends on some variable tending towards zero: convergent series, limits, continuity, differentiation and integration. It is the mathematics of arbitrarily small neighbourhoods, and it reached maturity in the nineteenth century. On the other hand, the ‘new’ mathematics of Weyl’s distinction is ‘global’; it studies the structures of ‘wholes’. Algebraic topology, for example, considers the properties of an entire surface rather than aggregations of neighbourhoods. Lautman re-draws the distinction:

In contrast to the analysis of the continuous and the infinite, algebraic structures clearly have a finite and discontinuous aspect. Though the elements of a group, field or algebra (in the restricted sense of the word) may be infinite, the methods of modern algebra usually consist in dividing these elements into equivalence classes, the number of which is, in most applications, finite.

In his other major thesis, (Essay on the notions of structure and existence in mathematics), Lautman gives his dialectical thought a more philosophical and polemical expression. His thesis is composed of ‘structural schemas’ and ‘origination schemas’ The three structural schemas are: local/global, intrinsic properties/induced properties and the ‘ascent to the absolute’. The first two of these three schemas close to Lautman’s ‘unity’ thesis. The ‘ascent to the absolute’ is a different sort of pattern; it involves a progress from mathematical objects that are in some sense ‘imperfect’, towards an object that is ‘perfect’ or ‘absolute’. His two mathematical examples of this ‘ascent’ are: class field theory, which ‘ascends’ towards the absolute class field, and the covering surfaces of a given surface, which ‘ascend’ towards a simply-connected universal covering surface. In each case, there is a corresponding sequence of nested subgroups, which induces a ‘stepladder’ structure on the ‘ascent’. This dialectical pattern is rather different to the others. The earlier examples were of pairs of notions (finite/infinite, local/global, etc.) and neither member of any pair was inferior to the other. Lautman argues that on some occasions, finite mathematics offers insight into infinite mathematics. In mathematics, the finite is not a somehow imperfect version of the infinite. Similarly, the ‘local’ mathematics of analysis may depend for its foundations on ‘global’ topology, but the former is not a botched or somehow inadequate version of the latter. Lautman introduces the section on the ‘ascent to the absolute’ by rehearsing Descartes’s argument that his own imperfections lead him to recognise the existence of a perfect being (God). Man (for Descartes) is not the dialectical opposite of or alternative to God; rather, man is an imperfect image of his creator. In a similar movement of thought, according to Lautman, reflection on ‘imperfect’ class fields and covering surfaces leads mathematicians up to ‘perfect’, ‘absolute’ class fields and covering surfaces respectively.

Albert Lautman Dialectics in mathematics

Discontinuous Reality. Thought of the Day 61.0

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Convention is an invention that plays a distinctive role in Poincaré’s philosophy of science. In terms of how they contribute to the framework of science, conventions are not empirical. They are presupposed in certain empirical tests, so they are (relatively) isolated from doubt. Yet they are not pure stipulations, or analytic, since conventional choices are guided by, and modified in the light of, experience. Finally they have a different character from genuine mathematical intuitions, which provide a fixed, a priori synthetic foundation for mathematics. Conventions are thus distinct from the synthetic a posteriori (empirical), the synthetic a priori and the analytic a priori.

The importance of Poincaré’s invention lies in the recognition of a new category of proposition and its centrality in scientific judgment. This is more important than the special place Poincaré gives Euclidean geometry. Nevertheless, it’s possible to accommodate some of what he says about the priority of Euclidean geometry with the use of non-Euclidean geometry in science, including the inapplicability of any geometry of constant curvature in physical theories of global space. Poincaré’s insistence on Euclidean geometry is based on criteria of simplicity and convenience. But these criteria surely entail that if giving up Euclidean geometry somehow results in an overall gain in simplicity then that would be condoned by conventionalism.

The a priori conditions on geometry – in particular the group concept, and the hypothesis of rigid body motion it encourages – might seem a lingering obstacle to a more flexible attitude towards applied geometry, or an empirical approach to physical space. However, just as the apriority of the intuitive continuum does not restrict physical theories to the continuous; so the apriority of the group concept does not mean that all possible theories of space must allow free mobility. This, too, can be “corrected”, or overruled, by new theories and new data, just as, Poincaré comes to admit, the new quantum theory might overrule our intuitive assumption that nature is continuous. That is, he acknowledges that reality might actually be discontinuous – despite the apriority of the intuitive continuum.

Weil Conjectures. Note Quote.

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Solving Diophantine equations, that is giving integer solutions to polynomials, is often unapproachably difficult. Weil describes one strategy in a letter to his sister, the philosopher Simone Weil: Look for solutions in richer fields than the rationals, perhaps fields of rational functions over the complex numbers. But these are quite different from the integers:

We would be badly blocked if there were no bridge between the two. And voilà god carries the day against the devil: this bridge exists; it is the theory of algebraic function fields over a finite field of constants.

A solution modulo 5 to a polynomial P(X,Y,..Z) is a list of integers X,Y,..Z making the value P(X,Y,..Z) divisible by 5, or in other words equal to 0 modulo 5. For example, X2 + Y2 − 3 has no integer solutions. That is clear since X and Y would both have to be 0 or ±1, to keep their squares below 3, and no combination of those works. But it has solutions modulo 5 since, among others, 32 + 32 − 3 = 15 is divisible by 5. Solutions modulo a given prime p are easier to find than integer solutions and they amount to the same thing as solutions in the finite field of integers modulo p.

To see if a list of polynomial equations Pi(X, Y, ..Z) = 0 have a solution modulo p we need only check p different values for each variable. Even if p is impractically large, equations are more manageable modulo p. Going farther, we might look at equations modulo p, but allow some irrationals, and ask how the number of solutions grows as we allow irrationals of higher and higher degree—roots of quadratic polynomials, roots of cubic polynomials, and so on. This is looking for solutions in all finite fields, as in Weil’s letter.

The key technical points about finite fields are: For each prime number p, the field of integers modulo p form a field, written Fp. For each natural number r > 0 there is (up to isomorphism) just one field with pr elements, written as Fpr or as Fq with q = pr. This comes from Fp by adjoining the roots of a degree r polynomial. These are all the finite fields. Trivially, then, for any natural number s > 0 there is just one field with qs elements, namely Fp(r+s) which we may write Fqs. The union for all r of the Fpr is the algebraic closure Fp. By Galois theory, roots for polynomials in Fpr, are fixed points for the r-th iterate of the Frobenius morphism, that is for the map taking each x ∈ Fp to xpr.

Take any good n-dimensional algebraic space (any smooth projective variety of dimension n) defined by integer polynomials on a finite field Fq. For each s ∈ N, let Ns be the number of points defined on the extension field F(qs). Define the zeta function Z(t) as an exponential using a formal variable t:

Z(t) = exp(∑s=1Nsts/s)

The first Weil conjecture says Z(t) is a rational function:

Z(t) = P(t)/Q(t)

for integer polynomials P(t) and Q(t). This is a strong constraint on the numbers of solutions Ns. It means there are complex algebraic numbers a1 . . . ai and b1 . . . bj such that

Ns =(as1 +…+ asi) − (bs1 +…+ bsj)

And each algebraic conjugate of an a (resp. b) also an a (resp. b).

The second conjecture is a functional equation:

Z(1/qnt) = ± qnE/2tEZ(t)

This says the operation x → qn/x permutes the a’s (resp. the b’s).The third is a Riemann Hypothesis

Z(t) = (P1(t)P3(t) · · · P2n−1(t))/(P0(t)P2(t) · · · P2n(t))

where each Pk is an integer polynomial with all roots of absolute value q−k/2. That means each a has absolute value qk for some 0 ≤ k ≤ n. Each b has absolute value q(2k−1)/2 for some 0 ≤ k ≤ n.

Over it all is the conjectured link to topology. Let B0, B1, . . . B2n be the Betti numbers of the complex manifold defined by the same polynomials. That is, each Bk gives the number of k-dimensional holes or handles on the continuous space of complex number solutions to the equations. And recall an algebraically n-dimensional complex manifold is topologically 2n-dimensional. Then each Pk has degree Bk. And E is the Euler number of the manifold, the alternating sum

k=02n (−1)kBk

On its face the topology of a continuous manifold is worlds apart from arithmetic over finite fields. But the topology of this manifold tells how many a’s and b’s there are with each absolute value. This implies useful numerical approximations to the numbers of roots Ns. Special cases of these conjectures, with aspects of the topology, were proved before Weil, and he proved more. All dealt with curves (1-dimensional) or hypersurfaces (defined by a single polynomial).

Weil presented the topology as motivating the conjectures for higher dimensional varieties. He especially pointed out how the whole series of conjectures would follow quickly if we could treat the spaces of finite field solutions as topological manifolds. The topological strategy was powerfully seductive but seriously remote from existing tools. Weil’s arithmetic spaces were not even precisely defined. To all appearances they would be finite or (over the algebraic closures of the finite fields) countable and so everywhere discontinuous. Topological manifold methods could hardly apply.