New entry history of mathematics and a couple of minor changes at philosophy.

]]>Thomas Holder has been working on *Aufhebung*. I have edited the formatting a little (added hyperlinks and more Definition-environments, added another subsection header and some more cross-references, cross-linked with *duality of opposites*).

added to *modality* a minimum of pointers to the meaning in philosophy (Kant).

New (tagged philosophy) entry structuralism, very stubby so far.

]]>Russian-speaking Philosophical Conference in Telegram!

Welcome here!

link -> Ph confa

]]>Several recent updates to literature at philosophy, the latest being

- Mikhail Gromov,
*Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2.*, pdf;*Structures, Learning and Ergosystems: Chapters 1-4, 6*(2011) pdf

which is more into cognition and language problem, but still very relevant, and by a top mathematician. As these 2 are still manuscripts I put them under articles, though I should eventually classify those as books…

]]>created *first law of thought*

started *universal exceptionalism*

related discussion is taking place on g+ here

]]>Hello all! I thought it would be a good idea to write an exposition of the current state of the formalization of Hegel, hiding as much mathematics as possible, so I did it! I mainly oriented myself after the dictionary you’ve set up here:

Since this is supposed to be an exposition of work on the nlab I thought I would give you a chance to take a look at it before I try to get it published somewhere (preferably in a philosophy magazine, to get them to pay attention), in case you have any comments.

I wasn’t sure how to best share the article, so I put it on Github:

https://github.com/nameiwillforget/hegel-in-mathematics

I also have some questions to completely finish the article:

In general categories, if we understand unities of opposites as adjoint modalities and define an Aufhebung of a unity $U$ as a unity $V$ such that one of its opposites contains the entire unity of the first Aufhebung, is it possible that the other opposite does not contain any of the opposites of $U$?

Is the shape of codiscrete types necessarily trivial: $\Pi\sharp\simeq *$? The example that is always given is that of codiscrete spaces, and I’ve written it from that perspective. Additionally, I feel like I don’t completely get the interplay between $\sharp=loc_{\neg\neg}$ and the logical understanding of types as propositions. I guess localizing the double negation is not the same operation as double-negating every type? What exactly does it do to the constituents of a type, if they are understood as the truths of a proposition? Or what exactly does the localization to Euclidean-topological $\infty$-gloupoids as a concrete example? I understand that it is the composite of a global section and an embedding, but how exactly does the embedding look?

Is the fermionic modality $\rightrightarrows$ or $\overline{\rightsquigarrow}$? On the Science of Logic page, it defines it as $\overline{\rightsquigarrow}$ at the beginning, but later on, and also in other nlab-articles and I think in dcct, it defines it as $\rightrightarrows$.

Who else should I mention as having worked on the formalization? As it stands, I’ve only mentioned Lawvere and Urs Schreiber.

Thanks for all the insights you’re keeping publicly available!

P.S. There clearly is some potential interest: https://www.reddit.com/r/askphilosophy/comments/q2w6ua/lawvere_and_hegel_use_or_abuse/

]]>Reasoning in mathematics is simple and subject to automation and discipline/system, because every concept (e.g. integer number, real number, derivative, integral, differential equation and its solution, etc.) can be expressed using some very small set of simple notions. If one considers the type theory approach to the fundamentals of mathematics, then there are only two basic types (entity and Boolean-truth) and all the other types, all the other notions and concepts are formed from those two simple types. Reasoning in mathematics is systematic because we completely know the content of the every concept. Yes, sometimes we imagine some new concepts (poetics of math) but even in such cases we manage to write down those concepts (or approximations of them) into the other concepts that can be traced to the first principles. Concepts in the mathematics are formed (or at least - can be expressed) in the bottom-up manner.

Reasoning about physics, about real world (ontology, metaphysics, nature, social world, humanities, emotions, mind, etc.) is very hard, because we can only make guesses about the eventual concepts, about the connections with other concepts and we don’t know the full content of the concept, every research discover new shades of some concept, concepts are created, merged etc. And all this happens in non-rigorous manner, because we don’t know the complete content of the concepts expressed in the first principles. We even don’t know the first principles that can be used for the real world.

Semantics of the natural language is perfect example for efforts to discover such first principles. E.g. reading from https://edinburghuniversitypress.com/book-elements-of-formal-semantics.html one can see the table that expresses each grammatical category as the derived type that is made from just two basic types:

Abstract type F-type S-type NP→S intransitive verb ff et NP→(NP→S) transitive verb f(f f ) e(et) A→(NP→S) be copula f(f f ) (et)(et) A→A adjective modifier f f (et)(et) S→(S→S) sentence coordinator f(f f ) t(tt) A→(A→A) adjective coordinator f(f f ) (et)((et)(et)) (NP→S)→S quantified noun phrase (f f )f (et)t N→((NP→S)→S) determiner f((f f )f ) (et)((et)t) (NP→S)→(N→N) relative pronoun (f f )(f f ) (et)((et)(et))

One can guess - if mathematics is the model of the real world, then we already have all the first principles, we just need more efforts to express such concepts as ’happines according to Aristotle’, ’ontology according to Hegel’, ’ontology according to British encyclopedia’, ’ontology according to some famous philosopher N.’ (we should always take into account that well defined concepts are connected to some personality in whose inner semantic we they can be found and only from such personal concepts the conventional concepts can emerge by convention in some scientific community, legal system, etc.) using the basic notions of math.

OK, I know that my thoughts are very childish. That is why my real question is this - is there some discipline in philosophy that tries to express the content of the each concept in some basic notions, is there discipline of the philosophy that tries to uncover such basic notions and types (be they the already known mathematical notions and types or something other)? What are the names of such disciplines of philosophy? What are common terms and research themes in such disciplines? Just keywords and names? Everything other I can find further myself.

I know, that there is metaphysical ontology (as opposite to applied ontology) but I don’t know the efforts to find the content of concepts and the first principles. I know that there is mereology, but it is about parts, about structures and systems, but the essence of the concept is something more that just its structural build-up. So, I am completely lost and I don’t know where to search further.

p.s. Why I am asking this? Well, I have zero internal/personal drive to understand world in such basic terms. I am just trying to automate thinking/reasoning (artificial general intelligence) for applied purposes and that is why I need systematic, disciplined, extensible and automatable way of handling concepts and I am just seeking for theories that are already created for such handling of concepts. Of course, they can not give the final answers, but they can be good starting point and the system can discover further horizons itself.

]]>I have added the quote on the “scientific world conception” of the Vienna Circle that David gave in another thread (here) to the entry *Vienna Circle*

I ran across your site as i was finishing my book/Art project and found your philosophy of interest as resonating with my own.

I reach out here to introduce myself and project - below please find the abstract and the book which is freely observable on line

Many thanks

John

3.0 i: Abstract/Letter of Introduction

Greetings fellow sojourner seeking,

Please pardon my interruption; I wanted to take the

opportunity to briefly introduce myself through my art

and, as such, invite you to peruse my book project with

title: i, in the palm of my mind : the chapters are

available to observe and consider online at

www.eidolononesuch.com

In my book, I explore - through various writing and

art formats - a series of desperate tho related concepts

ranging from cosmology to consciousness and free will.

And I propose and explore some intriguing models

you may want to consider such as regarding time and

how it results in the apparent EPR paradox and, as well,

a structural and mechanistic model for how matter and

mass connect as well as provide some detail on the

relationships of time/gravity and other forces.

Specifically,

I propose time as a 3-dimensional quasicrystalline net

of collapse events against the fabric of a fully collapsing

modular host pattern and describe where said pattern resides

I describe how to extract time from the pattern and evolve

a universe such as ours from a point containing

modular maths such as elliptics

I propose a unique model for the electron

(and nucleon components) as a resonant folded

wave/anti-wave (4:1 ratio in the electron) pattern in

the motif of a tetrix: a tetrahedral pyramidal fractal fold

of mass/void and the consequences of such regarding

the observed matter dominance in our universe and

its evolution

I describe the w/~w basis of the 3-state equilibrium

of photon/electron interaction as the tetrix tetrahedral

fold equilibrates with a cubic fold allowing point collapse

of part of the electron folded wave and how this yields time

and the EPR/Bell's inequality observations as waves and

anti-waves transition through the collapse state and

invert to yield the changed state

I explore how neutrinos may be the mechanism of

motion/momentum for the folded-matter waves and

how motion may be realized on a fully collapsing,

modular host with implications for how gravity

differs fundamentally from other forces

I elucidate a system - derived from neutrino oscillation

- detailing how rotational inertia is related and maintained

relative to linear momentum in a universe where the only force

is simplification and describe how momentum is prone to condense

with the greater frame of reference

I describe how the universe relies on de-construction

of its core information pattern, where the 'particle' machines

ur-data is extracted, sending the information by two mechanisms

to distant location and re-combining to yield the collapsible cubic

architecture: this results in alteration, temporarily, of the pattern

that consequently inflates our universe and this process can

occur reversibly without wave inversion beneath the

time collapse events

I describe entanglement as superposition/co-collapse

of mirrored, adjacent waves thus creating linkage to

subsequent collapse with inversion required as

observed time events

I propose straight-forward bases for the dark matter

and dark energy observations and describe the cycling

of universe iterations at our local site in the uberverse

including provenance of the high negative entropy

that is energy

Those are in part I of the book's triptych; if you find

these topics of interest, there are others that I find even

more intriguing in the remainder:

For example, I describe a functional definition of soul and

point to cryptic components, such as our ur-brain memory:

a cross-generational repository/information system which

provides survival utility as a fundamental evolutionary tool

This is interpreted in relation to the concepts of free will/choice

And I introduce potential reasons to question whether digital as

fundamentally differing, with relevance, from IRL

Please enjoy as you wont - it is intended purely and simply as art

and as such, simply and purely to inspire your art

Many thanks for all your good works,

J.

* June/July 20/20, JSM ]]>

started 道德经, for the moment just in order to record one paragraph which I found strikingly translated by Yiao-Gang Wen, here on Physics.SE.

]]>I added a discussion of space in Kant’s Transcendental Aesthetics in Critique of Pure Reason.

By the way, the translation of the quote from Kant in the section “On Aristotelian logic” seem a bit strange: I think the original German sentence was “Begriffe aber beziehen sich als Prädikate möglicher Urtheile auf irgend einen noch unbestimmten Gegenstand” (“But conceptions, as predicates of possible judgements, relate to some representation of a yet undetermined object.”).

PS The automatic function to create this thread in the nforum did not word.

]]>created an entry *category of being*, for completeness.

A general remark: people often write that, unfortunately or not, Eilenberg-MacLane’s term “category” is not that of, say, Kant. But in fact if read this way here, following Lawvere, then the former is a good formalization of the latter, after all.

]]>While concentrating on Category Theory as applied to the Web, I have also been following French Philosopher of technology Bernard Stiegler, as he was supportive of the creation of the Philosophy of the Web conference 10 years ago.

Recently he has been looking at the concept of locality which he argues has been repressed mostly after Aristotle, being overtaken since Plato with the notion of the Universal (The world of ideas), transferred into the Philosophers notion of God. Those philosophers who have tried to bring Locality back, such as Heiddegger with his concept of Dasein (being here), or Japanese philosopher Kitaro Nishida concept of place as a departure point, have had problematic relations with the Axis powers during the second world war. Yet Stiegler believes that since Schroedinger’s development in What is Life? of the concept of negative entropy as what fights *locally* entropy, forces us, if we are to take this seriously, to give a central position to locality. In any case the Web and the Internet create tensions between local cultures due to its creating ” a topological space without any distances” (see interview of late Michelle Serres).
I’ll see if I can find a good English paper of Stiegler that makes these connections clearly.

Now the Topos meant place in Ancient Greek (or see also Topoi on Topos: The Development of Aristotle’s Concept of Place). So I was wondering if people here who had deeper intutions about Category Theory than me, can see some insights that Category Theory can bring on these topics. Of course I bring this up here as Topos is also an essential concept in Category Theory which I believe one can summarise as the allowing one to connect logic and topology.

Perhaps modal logic captures this relation to location better. David Corefield in his published Chapter 4 on Modal HoTT writes of adjoint modalities:

]]>Choosing q equal to p, we see that a proposition sits between the images of the two operators (◻︎, p, ◇):

- necessarily true, true, possibly true following the pattern of
- everywhere, here, somewhere.

created an entry *beable*

(Surprisingly, this keyword does not have a Wikipedia entry…)

]]>Hi all, I added an entry for Leibniz’s identity of indiscernibles.

I thought it would be nice to include a discussion of the way in which univalence refines the identity of indiscernibles, but I am just beginning to learn about UF and was not sure what to say.

In Awodey’s 2013 exposition of univalence he states:

“Rather than viewing it as identifying equivalent objects, and thus collapsing distinct objects, it is more useful to regard it as expanding the notion of identity to that of equivalence. For mathematical purposes, this is the sharpest notion of identity available; the question whether two equivalent mathematical objects are “really” identical in some stronger, non-logical sense, is thus outside of mathematics.”

Thus we can regard univalence as “loosening” the notion of identity in such a way that it validates Leibniz’s law.

Certainly, I can just add something like this. However, it seems to me that a philosophical notion of equality that accords with intuition may profitably make a distinction between isomorphic structures when they are not “really” equal ontologically. Awodey is making a stance about “mathematical” rather than “real” identity, as is considered in (some formulations?) of Leibniz’s Law.

Is it possible to think about the connection to Leibniz’s Law like this:

Equivalence is akin to and entails a kind of “observational equivalence”/indiscernibility.

Thus we have the following putative principles:

- Indiscernibility of Identicals–intuitively true
- Identity of indiscernibles–controversial
- Indiscernibility is indiscernible from identity–intuitively true, and maintains some of the intent of identity of indiscernibles

So can we think of Univalence as a refinement that says “Indiscernibility is indiscernible from identity, so we may as well treat indiscernibles as equal by transporting isomorphisms into identifications.”? Or is this not the way to think about this/do folks in UF have a stronger stance such as “real” equality being absurd?

Best, Colin

]]>In “Natural duality, Modality, and Coalgebras”, in his thesis Meaning and Duality - From Categorical Logic to Quantum Physics, and elsewhere Yoshihiro Maruyama talks about $\mathrm{ISP}$, $\mathrm{ISP}$(M), how it is composed of $\mathrm{I}$, $\mathrm{S}$ and $\mathrm{P}$, but I can’t figure out what these stand for.

]]>A bit of a long shot, but since there is a Hegel page on ncatlab, I thought it would be interesting to see if there are any thoughts on the very interesting way Ruth Garett Millikan uses the word Category in her 1984 Book Language, Thought and Other Biological Categories. Btw, this can be thought of as a biological extension of Game Theoretical work by David Lewis on Language, such in his Languages and Language. There are categories of games, and even as coalgebras, so why not categories that would allow one to think of the biological, understood as that which reproduces.

Her departure point is that what seperates the physcial from the biological is the concept of reproduction. She starts with an example of a photocopier. Paper comes in, a copy comes out. There is clearly something that is the same from the input and the output, but that is not that the atoms are the same. This she argues can only be understood by taking the function of the photocopier into account. Its purpose is to make patterns of ink on a new paper that match those on the input, and other things can vary: eg. paper quality, color, etc… need to be the same.

With devices this function is (for us) easy to ascertain. For living things though as there is no manual for each species we need to work out what is the case by observing and theorizing. Here statistics by itself won’t do: she uses as example that most sperm never fertilise an egg. One has to look at the role the organ plays in the evolutionary survival of the organism, even the prehistory of that organism.

She then goes on to show how language fits then into that category, as words are reproduced from speaker to listener, and their function she argues is referential correctness. But not always: only at key points. So a single person could take a massive computer hard drive and fill it with “2+43=3” and that would not render arithmetic as we understand it nonsense, by overwhelming all existing literature on the subject with falsities. This is already very enlightening in so far as it I think correctly highlights the importance of etymology in the meaning of a word.

]]>Person page on a Turing recipient having a major work on causality, to record the references.

]]>Late last night I was reading in *Science of Logic* vol 1, “The objective logic”.

I see that the idea of cohesion is pretty explicit there, not in the first section of the first book (*Determinateness*, which has the discussion of “being and becoming” that Lawvere is alluding to in the Como preface) but in the second section of the first book, “The magnitude”.

There the discussion is all about how the continuous is made up from discrete points with “repulsion” to prevent them from collapsing to a single and with “attraction” that keeps them together nevertheless.

This “attraction” is clearly just the same idea as “cohesion”. One can play this a bit further and match Hegel’s *Raunen* to formal expressions involving the flat modality and the shape modality pretty well. I made some quick notes in the above entry.

On the other hand, that section 1 about being and becoming seems to be more about the underlying type system itself. Notably about the empty type and the unit type, I think

]]>I have added both to *proof* and to *experiment* pointer to

- Arthur Jaffe, Frank Quinn,
*“Theoretical Mathematics”: Towards a cultural synthesis of mathematics and theoretical physics*, Bulletin of the AMS, Volume 29,Number 1, July 1993 (arXiv:math/9307227)

with the quote (from p. 2):

]]>we claim that the role of rigorous proof in mathematics is functionally analogous to the role of experiment in the natural sciences

I added to principle of equivalence the following quote:

Either a thing has properties that nothing else has, in which case we can immediately use a description to distinguish it from the others and refer to it; or, on the other hand, there are several things that have the whole set of their properties in common, in which case it is quite impossible to indicate one of them. For if there is nothing to distinguish a thing, I cannot distinguish it, since otherwise it would be distinguished after all. (Tractatus §2.02331)

and a suggestive comment after it about considering such a statement for any given language, rather than the global setting stated in §1 (’the world’).

In the language given by the internal logic of a category one can never distinguish objects that share all their properties! My thesis is that ideas such as the internal language of a category and the Yoneda lemma have precursors in Tractatus, but I’ve not had time to sit down and nut out the details. Others have written about identifying the logic of the Tractatus (eg Potter’s The logic of the Tractatus, Weiss’ Logic in the Tractatus I: Definability, but I haven’t done a thorough search). I haven’t added these comments to the nLab anywhere, but I hope to do so when I flesh out my arguments.

In the process, I added to Tractatus Logico-Philosophicus some online sources for the text.

]]>created *The Music of the Spheres*, following Ravenel.