I have no problems with mathematical finance where a lot interesting math lies around though.

The most interesting portion of mathematical finance, I think, is where the interesting math that is lying around is not yet known to pure mathematicians. Elie Ayache (a paradoxically practical and theoretical person … an options trader with an enginering background / philosopher / ceo & founder of a financial software company) claims that - in his book The Blank Swan (not to be confused with Taleb’s The Black Swan) he has managed to “free the mathematics of contingency from the mediation of probability”: in philosophical form in an article like this The Turning (Wilmott Magazine) and in financial form in an article like this Actuarial Value vs Financial Price (Wilmott Magazine). (But, I should start a new thread rather than hijack this one if anyone is interested in generalized probability theory. The reason I brought this up is b/c we’re talking about mathematizing hegel in this thread and in other threads on the nforum, and perhaps Ayache is the hegel of the 21st century (in the sense that he is the only 21st century philosopher I am aware of who has written something that looks like it might (like Hegel’s Science of Logic) prefigure in conceptual form future mathematics). (There are actually 2 20th century books which have been called sequels to Hegel’s Science of Logic - Deleuze’s Cinema Books, where he elaborates a semiotics more advanced than Charles Sanders Peirce’s - but that’s another story. I just mention those books b/c if it is the case that they also prefigure math, it’s interesting that Hegel was able to do so with language, Deleuze (possibly) with cinema, and Ayache (possibly) with the actions of traders. Getting way ahead of things - what type of mathematics prefiguring medium could possibly come next?)

edit/addition: anyone who finds both philosophical and financial language alien, but still wants to see what a mathematics of contingency beyond probability can be can read Borges’ (very) short story Pierre Menard, Author of the Quixote. The terrain that distinguishes Menard’s Quixote from Cervantes’ Quixote is the terrain that contingency captures and that probability is blind to.

]]>Right, that cube arises from there being two different kinds of infinitesimal directions. I suppose if one were to formalize a factorization then that would be accounted for by the base topos itself factoring over yet smaller base toposes.

But I don’t have a systematic way (yet) to produce this factorization from *just* one cohesive topos. What I know how to do and did discuss is that

a) starting with an inclusion $\mathbf{H}_{reduced}\hookrightarrow \mathbf{H}$ that exhibits differential cohesion to produce its cofiber $\mathbf{H} \longrightarrow \mathbf{H}_{inf}$ where $\mathbf{H}_{inf}$ has infinitesimal cohesion.

b) or conversely, starting with $\mathbf{H} \longrightarrow \mathbf{H}_{inf}$ produce $\mathbf{H}_{reduced}$.

What Thomas’ remarks eventually made me think of is that maybe given just $\mathbf{H}$, we might be able to find some $\mathbf{H}_{inf}$ as a “smallest non-pointlike” cohesive base topos over which $\mathbf{H}$ still sits (maybe in that if you could find any other non-trivial such base topos, it would also have to sit over $\mathbf{H}_{inf}$).

This is vaguely plausible from the geometric interpretation of “infinitesimal cohesion”/”quality types”. But I don’t see yet how to make it precise.

]]>the factorization of any cohesive geometric morphism into something following by a QT

Does the order matter? We spoke once of a ’cube’ of cohesion. It seemed like you could take out the maximal infinitesimal part first or the orthogonal part.

]]>Oh, I thought you were referring to an arthogonal factorization system. That might have been the answer to what I was after: the factorization of any cohesive geometric morphism into something followed by a QT.

]]>This is e.g. proposition 3 of TAC 2007 p.47:

If $\mathcal{E}$ over $\mathcal{S}$ is both sufficiently cohesive and a QT, then $\mathcal{S}$ is inconsistent.

In the 1989 graph example $\mathcal{S}^{\{0,1\}}$ this occurs as the nonconnectedness of $\Omega$ and in the 1991 literally infinitesimal spaces he comments on this as lack of enough connected objects.

I recall also a remark on a factorization result for subtoposes of Paré, Rosebrugh(!?), Wood from 1989 in an Australian Journal that he points regularly to and calls relevant for infinitesimal generation of a topos in the ’Como 2008 lectures’.

]]>Thomas, could you recall the orthogonality of QT with “sufficient cohesion” for me? Thanks.

]]>Off-topic, but the blog conceptualmathematics was moved to private some time ago, unfortunately. If this was run by Lawvere, I would hope he would make it open again, not least because I vaguely recall it hosted some hard-to-find papers of his.

]]>An interesting suggestion. I need to think to about it. As Lawvere has pointed out on *conceptualmathematics* QT over S organize themselves into a 2-category, so I would be somewhat surprised when it would be possible to characterize a single QT as universal, though it might hopefully be possible to characterize the whole 2-cat somehow.

In any way, one would have to consider the lattice of essential localizations and look through the Johnstone papers which characterizes QT over S. Another possibility is to use the orthogonality to sufficient cohesion or ’pure variation’. I intend to go through the literature on QT soon also in order to report the useful results for the entry.

A caveat at this quantity-quality contrast. I brought up it in the entry because i thought it suggestive to think in these terms about topological invariants for dynamical systems, but the real contrast is indeterminateness-quality. It seems better to think of them as minimally cohesive spaces. Probably also hopeful to look at the graph example in TAC 2007-Como 2008 with the respective ex/intensive qualities to understand how they pick up information from the cohesive graphs in the domain.

In the 1991 Como paper Lawvere suggests on pp.9-10 that the SDG-infinitesimal spaces are of dimension $\epsilon$ which I guess means they are supposed to be just above (below?) the ground level 0 . So perhaps QT are more generally a sort of atoms of the essential subtopos lattice but Johnstone would have surely remarked on this if true. I need more time to think this over.

]]>Hi Thomas,

one aspect that you are maybe alluding to and which I know better how to relate to is the idea of *factoring* cohesion.

By this I mean given a cohesive topos $\mathbf{H}$, then it may happen that we have another cohesive topos $\mathbf{H}_{inf}$ and a geometric morphism $\mathbf{H} \longrightarrow \mathbf{H}_{inf}$ which exhibits $\mathbf{H}$ as cohesive over $\mathbf{H}_{inf}$.

Since both $\mathbf{H}$ and $\mathbf{H}_{inf}$ are assumed to be cohesiver over the given base topos $\mathbf{B}$, this means that the absolute cohesion of $\mathbf{H}$ “factors” through that of $\mathbf{B}$

$\array{ \mathbf{H} &&\longrightarrow&& \mathbf{H}_{inf} \\ & \searrow && \swarrow \\ && \mathbf{B} } \,.$In such a situation it makes good sense to say that as we pass to the right along

$\mathbf{H} \longrightarrow \mathbf{H}_{inf} \longrightarrow \mathbf{B}$that we “lose quantitative information” and “retain only more qualitative information”. This is simply so by the standard interpretation of the cohesive modalities, as we pass to the right with the left adjoint for instance we remember of spaces only their connected components (or their homotopy types in the homotopy context) and so in the middle stage we have something in between the full “quantitative” information and the bare “qualitative” information of $\pi_0$ (or $\Pi_\infty$).

This would be a sense in which I would understand why “quality types” refer to “quality”.

Of course in the above $\mathbf{H}_{inf}$ need not be a quality type for this interpretation to work, but a quality type would be a special case of this formalization of “qualitative information”.

Might it be possible to show that $\mathbf{H}_{inf}$ being a quality type makes it somehow universal among non-degenerate such factorizations?? In other words, could we show that factoring cohesion through a quality type is a “smallest non-trivial factorization possible”? That would be useful to know.

]]>@49. Well, finance vs. CS, a delicate matter: a guy like the late Francis Crick who always looked for the big intellectual adventure took to biology after WW II and then to conciousness at the end of his career. In the first case his Pascalian wager was a huge sucess and in the second case less so, and in any case he probably could have earned more on Wallstreet than the Nobel prize money.

I have no problems with mathematical finance where a lot interesting math lies around though. I would probably view it in his current form as part of applied statistical physics as the blueprint of equilibrium models there since Walras are actually taken from thermodynamics. So the choice seems to be a matter of personal choice, in general, it seems to me a much safer bet to think on risk at Wallstreet, than otherwise. So I guess it all comes down to the question whether you are interested in risk from a theoretical side or from a practical side. By the way speaking of risk, there is a paper on categorical risk in the recent TAC here though I haven’t looked at it yet.

A last remark on QFT: the thing I particular find intriguing with Lawvere’s approach to cohesion is that permits to reason synthetically on such methods and models i.e. it very much cuts across from QFT to petty graph toposes. E.g. that neural networks are field theory in disguise and as such hopefully amenable to cohesive analysis is probably less important per se than that cohesive SDG permits to throw out the particular models used and to focus on the really important features of the models. This seems to me to be a prerequisite to tackle the real big question which is not how the brain works but why it works the way it does (the problem of transcendental deduction in Kantian terms).

]]>Same guy.

]]>And here as well? ;-)

]]>I am the same Trent, good eye.

]]>Trent, you aren’t by same chance the same as the Trent who asked a question at MathOverflow here?

]]>Sorry for the off topic question, but, Thomas - which do you think is of more interest for mathematicians*: the mind/brain or financial markets?

(I’m asking because - while physics is the field** with the richest connections to mathematics - I think it is important for mathematicians (if they are exploring fields outside of mathematics) to explore multiple fields so as to keep their intuition general and not tied to any one particular strata of reality. People like Petitot & Andree Ehresmann bring alot of interesting mathematics into cognitive science …. but there are also lots of physicists & mathematicians on wall st who bring interesting math into financial markets. If one is already studying physics, finance has the advantage that (going from physics to chemistry to biology to neuro science etc..) finance is even more removed from physics than cognitive science is (and if you get even more removed from physics than finance and economics … the terrain is too subjective/fuzzy to be brought into rigorous interaction with math). Cognitive Science allows one to attempt to understand the nature of consciousness, whereas finance allows one to attempt to understand the nature of risk and probability.

I have no idea whether this book is any good, but just because we’re talking about taking methods from QFT and applying them to cognition: here is a book which applies QFT to financial markets.)

*by for mathematicians, I mean for people who care about the “physical mathematics” approach to a discipline outside of math just as much as they care about the “mathematical physics” approach **outside of the formal sciences

]]>Well, to throw in ’qualia’ was probably not a very lucky choice of me, as they tend to wake up bats and other favorite beasts of philosophers i.e. they are related to the question ’what it feels to have certain perceptions’. I used it to point to the psychological dimension and potential of this cohesion story, so I prefer to constrain my remarks to ’quality’ with a subjective dimension in the following.

Before doing this let me point out there is a mathematical model for qualia by Balduzzi-Tononi (pdf) which was morphed into an approach using presheaves in a Bayesian framework by David Balduzzi (here) where at one point he uses a tangent space metaphor for prior update which somewhat suggests laws of (mental) motions.

Most broadly QT suggest *qualitative* analysis of space, when you consider QT as degenerate or simpler spaces which replace the original space by valuing ex/intensive quality functors there - vaguely a projection to lower ( =infinitesimal) dimensions. The Poincaré remark pointing in the direction that several partial views permit to reconstitute the original space.

The subjective dimension enters then primarily by the necessity of being an earthling that cannot grasp the original space. In the kinship paper Lawvere uses at some point the term of ’rational neglect’ for the homotopical contraction I think. Sure a reasonable thing to do for a being with finite ressources.

More optimistically, one would try to link the infinitesimal space to the minimal soil of concious awareness (sort of Weber-Fechner pyschophysics like) - this is suggested e.g. by Petitot’s phonetics to phonology collapse provided it can be accomodated in cohesion+QT: although the full signal is processed by the auditory tract what enters concious awareness pertains to discrete phonological features.

To conclude with some Hegelian musings: Hegel’s WdL heavily depends on Kant’s KdrV which contains a dialectics as well as intriguing passages on the intensive (?) quality of conciousness which probably goes back to at least the substance dualism in Descartes, in short Hegel actually draws on a rich tradition here.

When I recall correctly Hegel links ’quality’ to determinate being or ’bestimmtes dasein’ for which you could interpret the determinateness as the pointedness and the connectedness as the being, probably the collapse of the cylinder as negation of becoming - a having come into determinate (self-identical) being. Well, I guess a cohesive reading of Hegel would be a fruitful undertaking for Hegel experts and should pay close attention to the text of WdL even at the cost that it does not fit the categorical bill.

]]>Hi Thomas,

would it be fair to say in the entry,then, that it remains unclear why an ambidextrous adjoint cyclinder faithfully axiomatizes “qualia”?

By the way, I once followed the implicit suggestion of Lawvere’s of reading Hegel’s *Science of Logic* and trying to formalize each “opposite of dualities” mentioned there by an adjoint modality, starting with identifying “becoming : nothing $\dashv$ being” with $\emptyset \dashv \ast$, as first suggested in *Some Thoughts on the Future of Category Theory*.

Now Hegel talks about quality, of course. Reading through what he writes and trying to match available adjoint cyclinders as faithfully as possible with Hegel’s poetry, I did arrive at an identification of “quality” with certain cohesive structure. This is the content of the entry *Science of Logic* (scroll down a good bit to see the relevant diagrams).

Regarding that footnote: sure, looks good.

]]>1. Lawvere wrote a message to conceptualmathematics in about 2008 that qualities like 'red' are intended to be modeled by ex/intensive qualities in the technical sense. I doubt that this claim is entirely thought through. Nevertheless due to his epistemological realism Lawvere has thought a lot about such things and e.g. has a sort of categorical theory of induction (most explicitly sketched in the objective logic 1994 paper, and hinted at in last year's Lawvere-fest abstract); so such a suggestion should probably not be dismissed easily. also from a more down to earth view QT occured as qualities in the analysis of cohesive graph toposes.

2. An example of 'qualitative analysis' in the non-technical sense in cogntion was given by Jean Petitot, who derived the categorization of the phonic continuum into the discrete vowel space with methods from catastrophe theory, suggesting the jet space example that Lawvere mentions. In general, mathematical models of cognition often take the form of neural networks which more or less are a form of field theory and differential geometry. So the idea that methods for QFT provide synthetic methods for cognition is not entirely farfetched. The highly abstract form DG takes in cohesion I refer to as 'transcendental', meaning 'concerning the conditions of possibility of perception' , a Kantian term.

(to be continued, I have to run!) ]]>

Okay, thanks. I have added a footnote.

Regarding “qualia”: as I mentioned before, I happen to be somebody who does not understand why quality types are regarded as

a concept where ’qualia’, the transcendental role of space in cognition (what it means in general to organize data into a space) and extraction of qualitative information from such space are brought together

Could you explain this?

]]>In order to fulfill such hopes the concept has to be available to researcher in CS or philosophy. In my view a purely technical approach to the concept would be an obstacle to achieve this. Imagine a cognitive scientist reading through Johnstone 1996 - would s/he come up with the idea that this a potential contender for 'qualia' ? I have no doubt that topologists can deal with quintessential localizations without ever having heard of Lawvre's 2007 when they pop up in their practice, I have more doubts, that CS ever reaches n-category theory without passing through more tangible realms before.

If you like to link to HOTT where you see 'synthetic homotopy theory', feel free! Calling Lawvere's ideas 'sHOT' is meant to make more knowledgable people wonder how they might relate to homotopy theory: always connect! ]]>

May I read this as saying that you are against adding a comment and pointer to homotopy type theory where you mention “synthetic homotopy theory”?

To me it’s curious where you identify tender sprouts and where their perils.

]]>I also think that this broad, somewhat vague perspective stays close to Lawvere's POV who calls it a QT and not a quintessential localization, makes repeatedly the link to quality in colloquial sense and stresses the role of the Hurewicz cat. I have no problem with bringing out more clearly the very important points-pieces role in this although I am not quite sure whether this a felicitous intuition for every QT over extensive cats and for this reason I demoted it after the localization remark.

To sum up, the three intuitions about QT are intended to be suggestive and not pedagogical. The pedagogical intention would be to stick more closely to and make more explicit Lawvere's ideas 1-verbatim. Concerning HOT we probably disagree, although it is somewhat uncertain what the putative synthetic HOT of Lawvere actually amounts to. Nevertheless we should grant Lawvere that his seemingly untimely view turns out to contain deep insights into the concept of space and the role it plays in cognition. ]]>

Thanks for the detailed reply and for displaying such detailed knowledge of the history of the development of the axioms.

I am all for the pedagogical exposition, the “$\infty$“-signs at *infinitesimal cohesion* could just as well be omitted as far as the definition is concerned. In view of pedagogy I find that explaining that $\Pi \simeq \Gamma$ characterizes infinitesimal objects (those with a single point in each piece) is less mysterious than saying that this means that “only qualitative analysis” is possible for these ojects. In fact I don’t understand this remark on qualitative analysis!
Also I think right in the vein of avoiding $\infty$-toposes in motivating “quality types” that reference to the homotopy category is less elementary than reference to infinitesimal objects.

For these reasons I still think that if the aim is to explain the actual geometric meaning of “quality types”, then the example $Sh(FormalPoints)$ is the most basic one.

On the other hand of course if the aim is to record Lawvere’s way of arriving at these concepts, then clearly a different route is indicated.

With respect to this though I feel the urge to make an edit on the following, but I’ll refrain before hearing back from you:

Where in the new version of the entry you say that quality types were

intended as ingredients to a

synthetic homotopy theory

it’s hard not to notice that with homotopy type theory there has meanwhile arrived an actual synthetic homotopy theory which axiomatizes not just “the homotopy category” – which is well known to be pathological – but its actual homotopy theory. With all due respect to Lawvere, it seems therefore to me that the aim of characterizing homotopy theory via cohesion is dubious, and that instead it seems compelling to interpret cohesion inside synthetic homotopy theory.

]]>