Interesting analysis of the development of the theory of philosophical thought. Inspiring research approach. Yes, I believe that the principles and theories described in philosophy may be important for the description and interpretation of specific results of scientific research and an attempt to explain phenomena and processes, etc. for which objective, instrumental research methods have not yet been developed.

Best wishes

Thank you Arno Gorgels

It is not surprising to me that structures that you find in set theory appear also in some metaphysical considerations: The domain of formal ontology explores this connection.

There are other examples of world representations where appears an organization of the world that is reminiscent of set theory, as for instance Leibniz' monads.

I would like to explore such examples and to give reliable descriptions or analogies by which means categorical constructions are represented in world models. It could be through physical laws where you find such structures, but it could be as well in ontological "laws".

One requirement would be though that the description or analogy is meaningful and respects some common intellectual standards (communicability, intelligibility, provability, etc. ). I would also include Occam's principle only to use those premises that are necessary to prove the conclusion: This requires a purification work of the argument. It should be possible to make explicit and verify each step of the argument.

In the text you cite you address some considerations of religious belief, of numerology, and so on. These arguments rely in general on principles of a different kind for instance "tradition" and can very often not be made explicit. I would ask to respect a certain tractability what sort of argument is used. To give an example: Theology for instance can be considered as a scientific domain in so far theological standards are respected.

My suggestion is to limit deliberately for the moment the discussion to philosophy of science or metaphysics in order not to mix up domains without justification.

Dear Arno,

I would like to ask you a few questions:

In your post you speak of applying Cantors set theory of infinities on the vacuum. Why specifically "Cantors" set theory? There is also the most spread Zermelo-Frankel set theory for instance, (or even more recent ones.)

What do you mean by vacuum? And how do you relate objects of set theory to it?

Dear Arno Gorgels

I do not know enough about your model of universe. What do you mean by volume quanta? How are they defined and what properties do they have?

I have the impression that your universe in some sense can be represented by point sets. It would be an "absolute" representation independent of a subject who lives in this universe, or does the duality subject - object play a role?

Best regards,

Tom Krantz

"Theory", to be applied IN GENERAL (in every place this concept does legitimately "come up"), is NOT something so particularly definable to be understandable IN SUCH A GENERAL WAY (for the just-indicated general purposes). AND: Any [ supposed ] such definition would skew its rightful-nature and purpose in any particular application (something philosophy has done "in spades" for centuries -- completely ruining Psychology for the entire history of Psychology). WE CAN DEAL WITH PART OF GOOD THEORY: There IS a definable standard of empiricism, but while this is required and always present in any good, real science, this does not ever cover the full character of any given theory -- which depends on its particular content. If you want to discuss the part of a "theory" which is understandable, discuss and understand good empiricism in whatever of its particular applications -- and how THAT (good empiricism), itself, LOOKS different in different, other contexts where that word is used. Quit striving for bigger, grander, but wrongful definitions (non-agreeable, not clearly understandable) of that which CANNOT, BY HUMAN NATURE (if you like), be understood "generally" as a clear concept onto itself (i.e. not as a clearly meaningful concept by-itself). I say: "Goodbye, Philosophy" -- source of false "existences" and related EXTREME DUALISMS. Philosophy is largely destructive junk.

To understand the main thing about good empiricism (and to see how I try to rescue Psychology): see my recent ANSWER (of today) to Iliescu Dragos w/r to the question : https://www.researchgate.net/post/Simple_but_ignored_What_is_good_for_General_Artificial_Intelligence_is_good_for_Psychology_and_if_not_good_for_one_not_good_for_the_other

Dear Brad Jesness ,

Thank you for your statement. I agree with you that the formulation of my question embraces a large panel of different philosophies. The intention was to set a context (in which there is a certain continuation along history).

The setting of rationalist philosophy has in my opinion more facilities to integrate analogies with category theory, which is understood in the mathematical sense (It is presented for instance in Saunders MacLanes Books). I say analogies because category theory gives the constructs I speak of a precise sense, which has to be well understood in order to integrate a precise philosophy of the cosmos.

Rationalist philosophies are close when considered in a narrow time window (Descartes, Malbranche and Leibniz).

I do not claim that rationalism is more appropriate than empirism or other philosophies to explain the world and to give answers to the question of humanity. Psychological questions are very subtle and I am not sure category theory is easy to harmonize with.

To give an example of the notions I have in mind I give the following definition of a category:

It consists of objects written A, B, C, ... and morphisms witten u, v, w, ... going from one object to another.

Morphisms can be composed as long as the final object of the first morphism coincides with the inital object of the second morphism.

To write it symbolically: if u: A -> B and v: B -> C are morphisms, then uv: A -> C is a morphism. Each object has an identity morphism e.g. 1B : B -> B and which is such that u 1B = u : A -> B and 1B v = v : B -> C.

Finally composition of morphisms should be associative in the sense (uv)w = u(vw) for w: C -> D.

Example: Objects could be logical formulae (of propositional calculus) and morphisms logical entailment. Another example would be Sets and set mappings

A functor is a mapping from one category to another mapping objects to objects and morphisms to morphisms preserving composition and identity morphisms.

An example of a functor would be the following: Sending a formula to a set and a logical entailment to a set theoretical mapping.

Even if this sounds to belong to the domain of mathematics it is clear to me that the notion of category can be understood more generally. For instance ontology does make use of lattice theoretic structures which hide categories in the sense I defined.

(to be continued)

Dear Thomas Krantz

Indeed there has been a long history of asking basically bad questions (ones too broad to be meaningful or helpful, as I indicated above). Just because we (humans) CAN ask bad questions does not mean we should (or that something good can come of it). Why we act as if "if we can ask them they must be worth asking" is beyond me (similarly, as people "philosophize" they act as if a "thing"/kind of "thing" can be labeled it is something onto itself AND worth discussing that way). You can also shoot yourself in the foot but because we CAN does that mean it is worthwhile?

What you seem to largely describe in your last post is some logic (logic forms), applicable to many things, BUT like the Subject of Logic itself, it is so apparent (when used _and_ needed/useful) that it is nearly self-explanatory -- this making for no reason to have big questions SUPPOSEDLY based on such stuff, but the "stuff" noted (as it often is again and again) really just pointing up basically self-evident things. AND YET, from this, you seem to think this gives some relevance and import to BIG, BAD questions (which I have more-well characterized, in my last post, above). You also mention and/or somehow put together some basically random gobbledygook for no clear purpose (and thus having no beneficial meaning), but (again) that maybe seems to you (and perhaps to some readers) to imply something "super", and THAT being really what I see as just part of a likely illusion/delusion.

Once philosophy is out of the way and out of the picture, there is only ONLY empiricism, properly defined (SEE LINK in my last post) -- that which is the nature of all that is useful in ones' studies.

Thank you Sydney Ernest Grimm

Dear Brad Jesness ,

I have understood that you prefer empiricism as a philosophy.

Can you present arguments in favor of it and also how it relates to the topic of the present discussion? The topic of the discussion I launched is not to present rationalism as the only possible philosophy, nor to present any philosophy, but to discuss how concepts of mathematical category theory can be more generally understood in order to find analogues in epistemology or (meta-)physics.

I will give another example taken from ontology: Suppose you have a certain number of concepts e.g. grass blade, a dog, a tree, a human. One possible generalization would be a physical object, but in so far we are interested in its most "close" generalization, one could suggest a living being, which is a particular physical object. In terms of the analogy one would like to explore "living being" would be the limit in a certain ontological category.

Dear Arno Gorgels

thanks also, see you!

Dear Thomas Krantz

THE arguments for empiricism: that is ALL there is (in some very real and necessary sense) and it is obviously fundamental; only it will establish any real science which is based on ability to communicate well (with very high inter-observer reliabilities. i.e. agreeability, AND confirmability/disproveability ); ONLY good empiricism can be a secure or VALID (BY scientific definition) basis of science; empiricism is inherently necessarily CLEARLY tied to observation and science is absolutely always clearly tied to observation [(minimally: some KEY observation(s), at some KEY time(s), and that being clearly sufficient (in the main) to reliably find, and convincingly explain, some sequence of phenomenon described by the science)]. SCIENCE always seemingly requires NOTHING ELSE (though sometimes the observational proof may seem indirect -- but it is only that) , nothing else, but THIS empiricism (as described above): WHAT YOU SEE/SENSE. And, THAT has been/can be tested and has been shown sufficient in all established science. And, all this is just what immediately comes to mind (but some characterizations against philosophy (below) is also part of an appropriate argument).

Now, why I responded to your Question: because you said (quoting you): "Do you think these principles have their signification a) for science ..." What principles?; but, NO in any case. I consider philosophy ONLY a menace and providing nothing good, but MUCH bad * -- as I have outlined in many essays (in Questions and Answers) over the years (in the thread mentioned, below). Go see them via my Profile --> Research --> Answers.

Perhaps it should be up to you to basically indicate how science itself is wrong and (once again) try to have philosophy "nose in" to this hallowed sort of matter (and distort and skew it). There has been a thread on this for three years +, "Can philosophy help to innovate and develop scientific theory?", where pro-philosophy people cannot make a rational argument on any good and understandable basis for philosophy being ANY good for science, not one clear instance (as opposed to what may simply be "claims"). (You can read my essays there, from over the last 2+ years, providing excellent arguments against philosophy (totally) and fully pro-empiricism . Empiricism is much more than a preference, it is MY WORLD.)

P.S. I have absolutely nothing against appropriately established generalizations, always established, or to be established, via the empiricism, described above YET covering ALL reasonable or cogent generalizations WHICH ARE ACTUALLY TO BE USED (read that last phrase , the last several words, over and over again for "therapy"). I LOVE good generalizations and I use and make them . BUT:

"Meta-physics" and "spirituality" are simply synonyms for ignorance, superstition, and delusion (this is the best, most useful interpretation).

* FOOTNOTE: I fight philosophy whenever/wherever needed (e.g. : YOUR Question).

For why I am considered a major "philosopher", myself: For one "thing": Because I have answered all the questions and addressed all the major issues in my own clearly central and important Project, https://www.researchgate.net/project/Seeing-if-Analytic-Philosophers-can-help-with-bringing-attention-to-Core-Problems-in-Psychology-and-to-Specific-Core-Proposals-for-a-new-Approach

Dear Brad Jesness ,

I do not consider science as the only domain on human knowledge. The aim of philosophy is precisely to discuss the what science is about and what human can know and so on. Empiricism and Rationalism have both been overthrown by Immanuel Kant. Human developmental psychology has its own rules, for which you might argue that empiricism is best adapted, I do not discuss that point. The development of children, of social communication abilities etc. etc. is very important but not the subject of this discussion. And it is also not the only important scientific domain. Why don't you start a new discussion on educational purposes?

I don't know if there is an experimental metaphysics. But again the question is not to discuss what people believe, but to present a rational analysis how principles of category theory (please look up https://en.wikipedia.org/wiki/Category_theory to see what I mean) can be seen in a more general context, the context of metaphysics. If you abandon rationality you can forget about all exact sciences and also about a lot of other domains, even statistical psychology.

Could you please respect the topic of this discussion!

Dear Thomas Krantz

I shall now leave this thread (Question), as you wish. BUT:

I will here and now note that in response to " I do not consider science as the only domain on human knowledge. ", I would simply respond that any cogent concept or set of concepts pivotally involves EMPIRICISM (which I would argue is not a "choice" but THE reality): this is what you should be able to fully conform to. To hell with Kant and your statement of an impossible nature. Kant has "overthrown" NOTHING. (I will never even ever mention ANY philosopher's name, to keep the embarrassment hidden -- and not make any pseudo-topics of philosophy (which most all of your "topics" are) active in another's mind).

You say " The aim of philosophy is precisely to discuss the what science is about and what human can know and so on "; this is only a very largely LARGE empty claim, something parroted, and/or a delusion in the philosophers' minde. You can say this or "define this" (out of any appropriate context, as you are so prone to do and as the case now), but it is all "bunk".

My Psychology Theory is meant to be central to all Psychology (aka General Psychology) -- and it is. In contrast: You just have just expressed more of your falsehoods that come from ignorance and which yield the subsequent delusions.

I WILL NOW GO, and let you be just with "your kind" (aka ilk) -- that is your only chance to "win" with me, or be happy (you could never win in any other way and you know where to find me, if you care to try). Be happy with just yourselves, and as you want to be (blither, blather) -- philosophy is not directly lethal to us "out of this loop" anyway but has inhibited good science for centuries, which is related to the death or demise of many (mainly because it simply, but almost constantly, distracts from "the good"). Let's all hope this thread can find a real and understandable and clear CONTEXT (there will be signs if you have, and signs if you have not (

Good discussion, following

To introduce one more concept I would speak about the notion of monad and algebra for a monad.

For the definition please see https://en.wikipedia.org/wiki/Monad_(category_theory)

To give a few examples taken from mathematics consider (You might skip the examples if it's too technical):

A) Double dualization monads:

    i) The functor T associating to a logical (intuitionistic) formula its double negation gives rise to a monad

    ii) The functor T associating to a set E the set P(P(E)) of sets of parts of E gives rises to a monad

B) Algebraic monads

    i) The functor T associating to a natural number n the set of decompositions of this number into formal sums evaluating to n.

    i) The functor T associating to a set b the set underlying the (free) vector space generated by b

Algebras for these monads would be:

A) i) A law allowing to pass from the double negation of the intuitionistic formula A to A (respecting the axioms of algebras for a monad)

    ii) A mapping from P(P(E)) to E (respecting the axioms of algebras for a monad)

B) i) The natural evaluation mapping

     ii) An evaluation mapping of formal expressions in a vector space

Similarly in ontology one could consider the functor T associating to a concept all its possible instances.

A T-algebra would be a law allowing to pass from a set of instances of a concept to the concept itself (e.g. from all possible real "apples" to the concept "apple", or from all possible triangles to the concept "triangle").

You see that monads and algebras for a monad could have an importance for the scholastic discussion nominalism vs. realism.

Dear Arno Gorgels

Yes, please share!

Best regards,

Tom Krantz

A standing wave of what?

'A smaller amount of volume' of what?

A volume of what?

Is space the first principle? (Because if not, it cannot be 'the first limitation of everything.') Is space permanent or impermanent? (Because if it is impermanent, it cannot be the first principle.)

How does space create electromagnetic waves? Isn't it the other way around: magnetism creates (centrifugal, spatial component) and destroys (centripetal, counter-spatial component) space -- and it is the balance between the two that determines whether the space is upheld or not? Does the center of a magnet, the source of the spatial component of the magnetic field, occupy any space? Does the center of a black hole (in which the centripetal component of the magnetic field overcomes the centrifugal component) occupy any space?

I would like to add the following (general) aspect to the question concerning cosmos models:

The classical common "scientific" theory supposes everything is composed of atoms, mathematically described one would now say in a topos with a line object. This line object is in most physical theories the field of real or of complex numbers, which (for historical or empirical reasons?) are supposed to best represent reality.

One could as well choose any other convenient field. There are many possible geometries. The role of physics is to relate these models to experience.

If we stay at the level of pure reason we have for instance no indication if the world is built of atoms or not, if it is infinite or not, etc.

Immanuel Kant has in his antinomies of pure reason described this situation so to say (https://en.wikipedia.org/wiki/Kant%27s_antinomies).

Should we abandon the idea of a unique mathematical "model" of the world and say, all models are equally valuable to describe "reality"? Is mathematical category theory helpful to this respect?

Does 'dynamics' get us out the chicken and egg situation? The setting out point being just where you position yourself.... Or there isn't a ground zero as such. My adopted position appears to be that em creates space its just that anyone using it to generate a concept of space will find that light has two orientations to them and that each component tells a different story about space. The resulting dynamics occurs to us. It plays out as the perceptual structure underpinning the phenomenon of vision. This dynamics does not occur to our instrumentation which is why our conceptual models derived from them are a bit of a mess.

Dear Arno Gorgels

I think you presuppose a sort of mathematical realism. Does the fact that we can measure or observe things oblige us to believe in a mathematical structure of the whole world?

Can you give precisions what do you mean by holisticism? How is it related to what physicists would call "local" or "global" laws? I mean to describe what happens on a billard table it is in first approximation not necessary to know what happens far from the table, whereas gravitation acts at large scale.

Would a finite model not have truth as well: I mean one could represent the world by a more or less large though finite number of simple components for instance?

Also, what do you mean by: creational?

Two interesting references on the link between "philosophical" and "mathematical" categories:

• Frédéric PATRAS « Catégories et foncteurs », Dictionnaire d'Histoire et Philosophie des Sciences, Ed. D. Lecourt, P.U.F. Sept. 1999.
• F.W. LAWVERE, « Categories of space and of quantity », (Philosophical, epistemological and historical explorations) Echeverria J. ed., Berlin, Walter de Gruyter, 1992.

Dear Arno Gorgels

you should indicate the source. What do you disagree with?

I would say, W. Lawvere's personal possible indeptedness to Hegel's dialectics and Engel's world system does not have a direct impact on mathematical category theory for which simply reason guides the investigations. If one agrees with the project to try to understand categorical principles in a philosophical context one should not think that they are more adapted for these than for any other "rationalist" philosophy. This is a nice aspect to be discussed though.

Dear Arnold,

I gave the basic definitions of mathematical category theory in a former post.

It is a theory of mathematical, logical, geometrical etc. objects together with the morphisms that transform one such object in another. Possible applications are in linguistics, computer science, etc.

The best reference is S. Maclane's book, Categories for the working mathematician.

Topos theory, which refines category theory, can be seen as a generalization of category theory.

There are some attempts to apply topos theory to physics. In my opinion it is a good context for developing physical theories as well.

Elementary topoi have a precise definition and the name has been chosen to respond to this definition. Elementary topoi have geometrical as well as logical aspects.

Arno Gorgels

Another possible and may be more intuitive definition is: A topos is a cartesian closed category with equalizers and a subobject-classifier.

( see https://en.wikipedia.org/wiki/Cartesian_closed_category and https://en.wikipedia.org/wiki/Subobject_classifier )

Topos theory is to some extend technical. But you can cope well with some fundamental examples:

1) The category whose objects are sets and morphisms are set mappings

2) The category whose objects are graphs and morphisms are arc preserving mappings

3) "Many cartesian closed categories of geometric spaces" (sorry, this is not very precise)

The cartesian closedness relates elementary topoi to Brouwers intuitionistic (constructive) logic. ( see https://en.wikipedia.org/wiki/Intuitionistic_logic )

A very clear and good (but also inevitably technical) reference is:

Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic, A first introduction to topos theory.

Concerning the former "nice aspect to be discussed" here is a citation from F.W. LAWVERE, « Categories of space and of quantity », (p.16):

"It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc."

Remark: I would not too much stress the aspect of "dialectical" though.

Arno Gorgels

Arno Gorgels

1) In maths. A similar sentence in nature would need probably "strong epistemological requirements"

2) adherent in a topological sence means any neighbourhood of the point meets its set theoretical complement in the space.

3) If you want that a sequence admits as a limit the point, it has to have stronger cardinality than the countable.

4) In order to apply a mathematical model to the world you need to make explicit how it applies - say a physics.

Arno Gorgels

It is a variant of Zenon's paradox for higher cardinality if you like.

Adevărul/reprezentării sau teoria rep*j

„totul este reprezentare”,

sau filozofie când realitatea este doar un absolut al acestei reprezentări, sau

„realitate nu se poate reprezenta pe ea însăși ca realitate ci doar ca reprezentare, umbră sau iluzie”.

sau

„totul este doar reprezentare, umbră sau iluzie,

unde umbra este reprezentare realităţii ca natură iar iluzia ca iluzie/realitate din acest motiv putem spune limitat adevărul/reprezentării sau al filozofiei la adevărul/umbrelor sau adevărul/iluziei/realitate sau al realităţii/interpretate, reprezentate sau realitatea ca raportare la un sistem de referinţă sau entitate/univers ca sistem de referinţă. În variantele limitate la om şi umanitate acest adevăr devine,

„totul este reflectare umană”

unde în acest caz reprezentarea este absolutul acestei reflectări al realități şi invers

„totul este reprezentare”

iar reflectările noastre doar relativul ei. Reprezentarea realităţii este doar absolutul acestor reprezentări dincolo de reflectările noastre.

(Fragment din lucrarea Mic dicţionar/absolut/realtiv sau dialectic- Tudor Păroiu)

Filozofie

Ca reflectare/nedialectică „filozofie” este,

„FILOZOFÍE, filozofii, s. f. 1. Știință constituită dintr-un ansamblu închegat de noțiuni și idei, care interpretează și reflectă realitatea sub aspectele ei cele mai generale; concepție generală despre lume și viață. 2. Totalitatea concepțiilor și a principiilor metodologice care stau la baza unei discipline sau a unei științe. 3. (Rar) Atitudine (înțeleaptă) față de întâmplările vieții; mod specific de a privi problemele vieții. 4. (Fam.) Lucru greu de făcut, problemă greu de rezolvat. – Din ngr. philosophia, fr. philosophie. Sursa: Dicționarul explicativ al limbii române, ediția a II-a | Permalink”,

sau

„FILOZOFÍE s.f. 1. Formă a conștiinței sociale, constituind un sistem coerent de noțiuni și idei care reflectă realitatea sub aspectele ei cele mai generale; concepție generală despre lume și viață. ◊ Filozofie lingvistică = teorie neopozitivistă care reduce obiectul filozofiei la termenii limbajului. 2. Totalitatea principiilor metodologice care stau la baza unui anumit domeniu al științei. ◊ Filozofia culturii = disciplină teoretică care se preocupă de definirea genetică, structurală și funcțională a fenomenului culturii. 3. Comportare, fel de a reacționa al cuiva în fața unor lovituri, a unor nenorociri etc.; fel al cuiva de a privi lumea și viața. 4. (Fam.) Problemă greu de rezolvat. [Gen. -iei, var. filosofie s.f. / < lat., gr. philosophia < philein – a iubi, sophia – înțelepciune, cf. fr. philosophie]. Sursa: Dicționar de neologisme | Permalink”.

Ca reflectare/dialectică „filozofie” este,

„echilibrul/dialectic al simultaneităţii lumină/întuneric”

dacă vorbim de generalitatea filozofiei ca reprezentare a realităţii dincolo de organizarea ca simultaneitatea haos/sistem a entităților/univers. În cazul limitat la entităţi/univers ca reflectare „filozofie” este

„echilibrul/dialectic la simultaneităţii cunoaştere/necunoaștere”

ca reflectare a reprezentării lumină/întuneric față de sisteme de referinţă, indiferent de organizare deoarece şi nesistemele sau haosul pot fi sisteme de referinţă chiar dacă au convenții specifice (indiferență/întâmplare) fără legături dar cu siguranță au lumina şi întunericul lor indiferent de realitatea sau reprezentarea acesteia. Dacă limităm filozofia la lumea sistemică putem spune „filozofie/sistemică” este,

„echilibrul/dialectic al simultaneităţii cunoştere/necunoaștere/sistemică”

deoarece se referă doar la științe şi neștiințe sau fenomene limitate ale gândirii având în vedere că haosul nu admite nici cunoaştere dar nici necunoaștere/sistemică neavând reguli, legături, etc. şi știind că dialectica/filozofiei este simultaneitatea lumină/întuneric care limitată la sisteme este doar cunoașterea/necunoaștere/sistemică a reflectării şi nu a realităţii sau reprezentării ei indiferent de natura reprezentării sau a realităţii reflectate în acelaşi timp şi ca reflectare simplă şi normată. Ceea ce ne duce la concluzia că reflectarea/sistemice este doar cunoaştere/necunoaștere/sistemică simultan. Înainte de a ne justifica ne permitem să atragem atenția asupra limitelor definițiilor clasice care nu se hotărăsc dacă filozofia este „Știință constituită dintr-un ansamblu închegat de noțiuni și idei, care interpretează și reflectă realitatea” sau „formă a conștiinței sociale, constituind un sistem coerent de noțiuni și idei care reflectă realitatea”. Aceste definiții ne arată confuzia sau limitarea lor față de realitatea ca natură a oricărei entităţi/univers şi realitatea ca spirit a acestora dar mai ales că noi sau alte entităţi/univers nu putem cuprinde nici realitatea nici reprezentarea ei ci doar reflectarea lor față de noi ca sisteme de referinţă. Diferenţa este clară una este realitate ca reprezentare şi alta reprezentarea raportată la un sistem de referinţă (noi sau alte entităţi/univers) ca reflectare dar şi această reflectare ca dialectică sau nedialectică a reflectării, indiferent de natura acestei realități. Dicționarele confundă reflectarea ca natură cu reflectarea ca spirit sau idei sau iluzie/realitate. Totodată confundând de cele mai multe ori realitatea cu reprezentarea ei în idei ca spirit chiar dacă şi ea este o reprezentare a realităţii mai ales că ambele variante nu sunt realități ci doar reflectări ale reprezentării realităţii. Nedialectic putem spune că filozofia este,

„știința/neștiința reflectării realităţii”

sau

„știința/neștiința reflectării comparațiilor realităţii”

sau,

„știința/neștiința reflectării rapoartelor cu realitatea”.

indiferent de natura lor științifică (rapoarte identice) sau neștiințifică (rapoarte neidentice) sau filozofică (simultaneitate) sau de natura elementelor comparate. Din păcate aceste definiții sunt aferente sistemelor şi gândirii sistemice limitate la ştiinţă şi neștiință în nici un caz dincolo de sisteme de orice natură. În plus cunoașterea şi necunoașterea nu însemnă doar ştiinţă ca dovadă că sentimentele, intuițiile sau instinctele noastre sau ale plantelor şi animalelor sunt în permanență cu noi şi ne încurcă sau ne ajută după caz. Să nu credem că ele sunt inconștiente este o limitare prea mare şi inutilă știind că inconștiența este o inexistență. Limitată la ştiinţă sau neștiință filozofia studiază rapoartele realităţii noastre (reprezentare/reflectată sau iluzie/realitate) cu realitatea în sine, cu natura care suntem şi ne înconjoară atâta timp cât la celelalte realități nu putem ajunge. Este forma de reflectare a unei reprezentări într-o altă realitate în cazul nostru reflectarea reprezentării ca natură, realitate în sine sub orice formă ar fi ea, în spiritul unei entități/univers, în particular om sau altceva. Poate fi o reflectare convenţională sau una neconvențională sau așa cum o redefinim noi dialectică sau nedialectică ca raportare al un sistem de referinţă. Chiar dacă noi nu cuprindem reprezentare putem extrapola limitatul ei reflectat. Filozofia este simultaneitatea convențiilor unei entităţi/univers, dincolo de natura realităţii sau reprezentării, ce sunt acumulate în memoria ei ca rezultat al experiențelor spiritului fiecărei entităţi/univers la nivel de formă, existență şi spirit. Adică filozofia cuprinde reflectarea realităţii ca generalitate sensibilă, intuitivă, rațională, etc. (reflectare generalizată spiritual) nu doar a gândirii sau cum se credea anterior doar reflectările rațiunii umane. Să nu limităm gândirea (capacitatea de a face rapoarte, simultaneități sau nu) la nivelul gândirii umane ci la gândirea oricărei entităţiunivers în funcție de componentele acesteia, conștienţă, conștiință, voință, logică, rațiune, etc. în noile interpretări absolut/relative atâta timp cât acestea sunt rezonanțe de o natură sau alta. La urma urmei gândirea omului este doar o convenție umană, o capacitate de a face comparații, rapoarte, etc. care ne permit reflectarea rereprezntărilor realităţii ca filozofie a spiritului fiecăruia dintre noi într-o formă sau alta. Şi natura are filozofia ei ca şi dincolo de limitele noastre sau a altor entităţi/univers ca absolut/realtiv atât sistemice cât şi haos. Am putea spune că existăm o dată cu filozofia şi prin filozofia fiecăruia dintre noi sau fiecărei entităţi/univers dar nu trebuie să uităm că dincolo de filozofie există realitatea şi reprezentările ei sistemice sau nesistemice chiar dacă doar ca reprezentare pentru noi sau doar intuiție. Să nu limităm filozofia doar la reprezentarea sistemică a realităţii şi haosul are filozofia lui dincolo de orice organizare. Orice entitate/univers are filozofia ei şi în acest context orice obiect, fenomen sau de altă natură are filozofie proprie din acest motiv putem constata că totul este filozofie sau reprezentare a realităţii, depinde doar de sistemul de referinţă la care se raportează filozofia acestora.

(Fragment din lucrarea Mic dicţionar/absolut/realtiv sau dialectic- Tudor Păroiu)

UgY��Q

I do not know English and the translations make it on Google, which anyone can do, especially as the translations become more accurate every day.

Truth / representation or reputation theory

"Everything is representation,"

or philosophy when reality is only an absolute of this representation, or

"Reality can not represent itself as reality but as representation, shadow or illusion".

or

"Everything is just representation, shadow or illusion,

where shadow is a representation of reality as nature and illusion as an illusion / reality for this reason we can say limited the truth / representation or philosophy to truth / shadows or truth / illusion / reality or reality / interpreted, represented or reality as a reference to a system of reference or entity / universe as a reference system. In variants limited to man and humanity, this truth becomes,

"Everything is human reflection"

where in this case the representation is the absolute of this reflection of reality and vice versa

"Everything is representation"

and our reflections only its relative. Representing reality is just the absolute of these representations beyond our reflections.

(Fragment from the Mic dictionary dictionary / absolute / real or dialectical - Tudor Păroiu)

Philosophy

As reflection / dialectical "philosophy" is,

"Philosophy, philosophy, s. 1. Science consisting of an ensemble of ideas and ideas that interprets and reflects reality in its most general aspects; general view of the world and life. 2. The totality of the conceptions and methodological principles that underlie a discipline or a science. 3. (Rarely) Attitude (wise) to the events of life; way to look at life issues. 4. (Fam.) Hard work, hard to solve. - From ngr. philosophia, fr. philosophie. Source: Explanatory Dictionary of Romanian Language, 2nd edition Permalink "

or

"FILOZOFÍE s.f. 1. Form of social consciousness, constituting a coherent system of notions and ideas that reflect reality under its most general aspects; general view of the world and life. ◊ Linguistic philosophy = neopozitivist theory that reduces the subject of philosophy to the terms of language. 2. The totality of the methodological principles that underlie a certain field of science. ◊ The philosophy of culture = theoretical discipline which is concerned with the genetic, structural and functional definition of the phenomenon of culture. 3. Behavior, how to react to someone in the face of blows, misfortunes, etc .; someone's way of looking at the world and life. 4. (Fam.) Problem hard to solve. [Gender. -which, var. philosophy s.f. /

Dear Tudor Păroiu ,

Thank you for the interesting text defining philosophy. It can contribute to the question on the role of logics in philosophy. You speak on intuition and representation of the reality. This is already philosophy of mind and knowledge.

How is logics related to representation of the world? This is a worthwhile discussion and has articulations with the philosophy of science we have been speaking about until now.

We should keep in mind the initial question and relate new aspects always to it.

Logics is about logical entailment (A=> B): From A follows(logically, causally, etc.) B; or about conjunction: From (A and B) it follows A and it follows B; or disjunction etc.

From A follows ( A=>B ) => B for any B but what about the reverse implication? For which B can it be true?

Philadelphia, PA

Dear Krantz & readers,

I sympathize with the worries concerning "total abstraction" (and excesses of formalism).

You are, of course correct that contemporary physics involves gigantic expansions of scope in that, say, experiments and results in the particle accelerators at CERN and elsewhere, are extrapolated in cosmology and theories concerned with the origins and expanse of the observable (even sometimes the un-observable) universe. Yet at the same time, it is assumed that things taking place at those extreme distances have no local effects on the outcomes of experiments. On the other hand, worries of this sort are put aside on the basis of considerations of scale and the open possibilities of further testing of general laws proposed or hypotheses on offer. That is the empiricist attitude, as I see it. (Aristotle's rigid, unchanging distinction between "essence" and "accident" is rejected. )

Regarding logic in particular, it is implausible to regard it as an experimental science. Still, since the time of Aristotle, logic has not only been expanded but also revised. It has been expanded, for instance, in the explicit logic of relations. It has been revised as concerns the existential suppositions of the Aristotelian universal propositions. Briefly, put, the Aristotelian universal proposition, "All A's are B's" was taken to imply that "There are A's." The modern universal proposition, "(Ax) (Ax only if Bx)" will count as true on the grounds that each instantiation has a false antecedent, i.e., if "not-(Ex) Ax" is true, then so is "(Ax) Ax only if Bx)"; if there are no unicorns, then "All unicorns have a single horn" is true.

As I say, Aristotelian logic proceeded on the assumption of "completed science." But given that we regards our sciences an incomplete and expanding, we make greater space for hypothetical introduction of concepts. The "proof" of this, would seem to be (at least partly) in the pudding of the factual facilitation of the expansions of the sciences.

H.G. Callaway

---you wrote---

Classical physics is very vulnerable since it extrapolates the results of local observations to unknown realms. Yet, for the time being there is nothing else to refer to. The problem of total abstraction may be that it takes undue distance from reality.

Dear Mr. Thomas Krantz,

I would like to tell you a few short ideas so I will not abuse your patience. First, mathematics uses identity and equality in its logic. Unfortunately, the anthura does not admit identity and I will give one example. Two atoms (generally many claim their identity) are never identical, provided that we do not limit our identity as mathematics does. To be identical means that all their paramters are identical, but the nature around them is identical because the entities / universe around them are differently distributed to them, and the influences of the outer universe become different from each atom. An atom in my chair can not be identical to one from your body never, on the one hand because it does not occupy the same space / time dimension on the other because the outer and inner universe of each are different and unevenly distributed. Another crucial argument of my theories is that in reality, but also in its reflections on different reference systems, there is a paradox that is no longer logical as Husserl's mathematics or pure logic says, plus to forget the chaos that exists indifferently why we believe or want to believe we or the geniuses of science. Mathematics and logic no matter who supports it has its limits and its relative as proof that it has countless exceptions, limitations or impossible (illogical) or indeterminate cases that can not be resolved by the mathematical or general philosophy of the past or present to our claims. a philosophy / dialectics (absolute / relative) that must continually take into account the space / time dimension of any entity / universe or science or ignorance. As a reality or just as a substance (as a matter of nature), "A" is not identical to "A" never (an A on a stone can not be identical to an A on a paper, etc.) and as hypothetical idea or our conventions , but there is much to talk about, I apologize.

Dear @H.G Callaway and readers,

The approach I suggest is to try to understand the world without mixing up our scientific representation with the world itself. I very much like the unbiased reflection of the early Greek thinkers.

To give an example: Is the world built up out of atoms? I do not doubt that you can find atoms, but is everything built out of atoms? I don’t know. I think it is a questionable philosophical principle.

A model for this is given by Stone spaces, giving examples of geometrical spaces that might contain very little points. It disturbs our conception of the world. Now certain theorems say that under certain hypotheses we can complete space so that it is built up out of atoms. But are these hypotheses met in our representation of the world or should we prefer such a representation etc.

Does logic apply to our world or is logic (and geometry also) only, as Kant might formulate it, in our minds?

(I reject excesses of formalism when it no longer is at the service of humanity.)

Dear Tudor Păroiu ,

I respect your analysis of equality and identity. Any two flowers are different, that's true, but it is also true that both are a flower.

If we say there is no flower but concrete examples of flowers, we adopt a form of realism(?)

As soon as we use a language of signs we give the possibility to identify items of mental representation even if physically we would be able to find some differences.

I think it is a necessary condition for being able to find a common language.

Philadelphia, PA

Dear Krantz & readers,

I think that Kant's conception of geometry as a priori pretty much fell with the spread and acceptance of the consequence of Einstein's GR. If non-Euclidean Geometry actually applies to physical spacetime, then Kant was wrong.

Logic applies to our theories of the world--including our best theories. Logic has to do with which statements follow from others given as premises. Logic, as I pointed out, has been revised, but not very often. We properly treat it with a good deal of theoretical conservatism.

I don't think we should be surprised at this conservatism, since logic has had a long steady development, from Aristotle onward.

H.G. Callaway

---you asked---

Does logic apply to our world or is logic (and geometry also) only, as Kant might formulate it, in our minds?

Dear H.G. Callaway ,

One could revise Kant by accepting that we have a plurality of geometrical models that we can use to represent the world. Euclidean Geometry, non Euclidian, and other geometries as well... Does this affect the conception of a reality behind it? I don't think so. (Using different instruments to perceive the world might give you different images, but is not in contradiction with the realist idea that they can be unified. Locally the earth is flat, seen from space it is a sphere, and both are right to some extend.)

Dear Mr. Thomas Krantz, let us not confuse the absolute with the relativity it contains, just as we do not have to confuse reality with its representations to which nobody and nothing reaches nor believe that the absolute does not exist or reality. Entities / universes or new ones are just systems of reference, and the reality we perceive is only a reality refined by our senses or, more precisely, its reflected representations, ie, referenced to reference systems. Just as we do not have to confuse the parts with the whole they form, even if the whole is always different from its parts as a whole. Unfortunately, we humans believe and believe our belly of the earth, what we are not, but we mostly use interpretations and symbols on Aristotle's eye, which we do not analyze carefully after 2000 years. I will give one example, but there are countless examples of our inability or limitations. Aristotle says matter is "the substance from which the objects are made," and unfortunately we have taken this definition, and we believe today, when we are in the quantum physics age, that the office we write is matter without thinking that it is in reality, a matter / energy substance consisting of fullness and gaps, but especially matter and enrgy which, in a limited way, we call it matter only.

Dear Arno Gorgels

We have been taught that I. Kant has reconciled the rationalist views of Descartes with those of the anglo-saxon empiricists (Locke, Hume, etc.) by localizing the rational structure in our minds. (In the Freudian language one would call this psychological mechanism projection I suppose.) One would have to have a closer look at the source I think. We have been taught also that I. Kant had done something similar as Copernicus when he placed the sun at the center of the universe an no longer the earth. Now we live in the conviction that also the sun is not any longer the center of the universe but it has no center, (and we get back to localize the center of the universe in ourselves.) If Kant was not absolutely right, he has some merits though (but certainly not the merit to have correctly interpreted Anselm's ontological proof).

What is then the relation between logics (some view as a sort of game), numbers and the world? To stress a bit Kant's interpretations: We see a unit in some things and other things separately. (This is at the source of the opposition of Parmenides and Heraclitus, right?) But is no objective view, it is always relative to a subjective standpoint. What forces uses to agree on views?

Philadelphia, PA

Dear Krantz & readers,

Actual unity within and among the sciences and scholarly disciplines is an occasional accomplishment, and not an a priori presupposition. Kant was wrong in claiming that space and time are a priori "forms of intuition" --though we can still be as conservative as the facts may warrant in the use of related concepts and their developments. What is more plausible is that something like Euclidean geometry is "built-in" to the visual systems of higher mammals (at least) which may offer some explanation of our difficulties in imagining, say, curved space time or four-dimensional objects, etc. That is an empirical hypothesis, concerning which one might seek evidence and experimental tests. It is not, certainly, a "presupposition of all possible experience."

We strive for comprehensive unity, just as we seek thematic unity in an essay on a topic. This is sometimes found, sometimes not. The danger in over-emphasis on unity is that it encourages false or superficial unity --sometimes, at the extreme, closed ideological systems of thought--which fail to stand up to examination or propose no testing or critical examination.

Nothing I have said here should be supposed to put an empirical, moderate realism in question--somewhat in the spirit of Aristotle. Just as formalism may proceed by exclusion of needed details, the pursuit of unity and comprehensive system may eventuate in closed, inbreed, metaphysical dogma. That is the danger in over-emphasis on unity.

It is always interesting when use of differing instruments and methods lead to the same or quite similar results. (Consider why it is that we have two eyes.) But when differing methods lead to conflicting results (as, say in recent determinations of the Hubble constant), that can be even more interesting.

H.G. Callaway

---you wrote---

One could revise Kant by accepting that we have a plurality of geometrical models that we can use to represent the world. Euclidean Geometry, non Euclidian, and other geometries as well... Does this affect the conception of a reality behind it? I don't think so. (Using different instruments to perceive the world might give you different images, but is not in contradiction with the realist idea that they can be unified. Locally the earth is flat, seen from space it is a sphere, and both are right to some extend.)

In my opinion category theory is fine as a taxonomy of mathematics, but there is asymmetry between functions and objects in category theory in that objects are defined by their functions, but not vice versa. Personally though I do not find definitions in category theory easy to follow. Compare the definition of "x is a natural number" in the second order predicate calculus as "for all properties P, [P(0) AND for all y P(y) IMPLIES P(s(y))] IMPLIES P(x)" for initial number 0 and s a one-to-one function such that for s(y) is not 0 for any y, with the definition at https://en.wikipedia.org/wiki/Natural_number_object.

As for the use of category theory in science, I can see it might be useful in providing a way of expressing a physical theory that would otherwise be difficult to express, in much the same way as Riemann's theory of manifolds supported the development of general relativity and Hilbert spaces provided a framework for quantum mechanics.

Philadelphia, PA

Dear Powell & readers,

How to you see the relationship of "classification in mathematics," or "classification in science" to Riemann's influence on GR or the role of Hilbert spaces in relation to QM?

It would seem that the mathematical systems have these uses however they may be classified. Right? How is "category theory" supposed to help?

H.G. Callaway

Dear Andrew Powell ,

It might also be a question of taste. Mathematical category theory has may-be sometimes not the most simple definitions, but very often quite good general ones. The definition of a monad and the corresponding algebras might be very little intuitive at first, but a posteriori, one can see that it is not only a generalization of groups, of many algebras, of topological closure operators etc. etc.

It is for sure not the only possible language, there is also second order predicate calculus (and other formal systems). But try to define an algebra in such a general form in it(?).

Dear H.G. Callaway and readers,

You are right, mathematics is often concerned with classifying mathematical objects. Category theory in contrary tries to make evident how different categories relate to each other. Adjunction is a very fundamental aspect and one of the basic principles in category theory is the fact that there is a correspondence between monads and adjunctions.

Now your question is very complex. One could also ask (with Aristote if you like) "What is physics?" or "What is natural science?".

Dear H.G. Callaway

Category theory is a rich taxonomy of mathematics, and it would be natural for physicists to find a category that fits the physics that they are trying to model. Einstein was greatly helped by the existence of Riemann's theory of manifolds (which was developed by others in the intervening 50 years), and quantum mechanics would not be so clear if it were only represented in the language of partial differential equations. I am not sure what the category or the physics would be though. Category theory has been influential in theoretical computer science and in computational linguistics.

Dear Thomas Krantz

I would not try to express all of algebra in the second order predicate calculus (or even in finite type theory). With finite type theory you could do it, but it would look cumbersome. Yes, category theory is good at rendering algebra, but then not all of mathematics is algebra. Geometry and general topology are not entirely algebraic. Rather there are algebras associated with the structures that mathematics studies, which enable a mathematician to characterise the structure. But there is power in new notation, whether it was Leibnitz's language for the differential calculus, Hamilton's quaternions or, my personal favourite, the language of geometric algebra (due in recent times to A. Lazenby and C. Doran).

Dear Andrew Powell ,

I would add also geometry and algebra. Consider the topic of sheaves, and Grothendieck's work.

I do not consider the main aim of category theory to classify mathematics. I would rather think it's role is to propose a general language and unifying principles for different domains like geometry, algebra, combinatorics, graph theory, partial differential equation theory, and so on.

Now Physicists are not used to it but rather stick to traditional formalisms as QM or semi-riemannian geometry. So it's important in my eyes to ask whether it is adequate to use a particular geometry rather than a general formalism (as Hilbert geometry rather than synthetic differential geometry to give an example).

Dear Thomas Krantz,

You are probably right that it is a matter of taste. I still think that category theory is mathematics as algebra, although you are quite correct that it came out of algebraic topology and differential geometry. A lot of number theory, analysis, probability and statistics does not naturally fit category theory, and I would not use category theory to derive theorems in Euclidean geometry (or indeed any constant-curvature geometry).

Dear Andrew Powell ,

Using categories for this is may be without precedence but certainly a good idea...

I will try to figure out the necessary axioms. Symmetric spaces are particular quotients of Groups, and groups are easy to bring in accordance with categories. Categories are in my opinion a good setting for this - much better than using the much too elaborate differential geometry and then a definition of curvature etc.

-- Quoting --

I would not use category theory to derive theorems in Euclidean geometry (or indeed any constant-curvature geometry).

I think that category theory could help us in understanding some concepts of "rationalist" philosophers like Leibniz. I mean esp. Leibnizian relational concepts of space and time. Interesting attempt, to build up an "arrow philosophy" is M. Heller's article DOI: 10.12775/LLP.2016.013 (in OpenAccess)

Thank you Paweł Polak , for the hint.

Best,

Tom Krantz

Dear Andrew Powell ,

In fact to give more details in addition to the preceding post a symmetric space is grosso-modo a Lie-group together with an involutive group automorphism.

References: https://en.wikipedia.org/wiki/Symmetric_space and J. A. Wolf, Spaces of constant curvature

Interesting comments and remarks

Dear Thomas Krantz

I agree with you that the notion of a symmetric (Riemann) space is a nice characterisation of geometric spaces of constant curvature. However, like the category-theoretic definition numbers, the characterisation of such geometric spaces requires some quite deep mathematics. In the case of natural numbers, the definition describes an isomorphism between two sets of natural number structures, while the involution property in the definition of symmetric space (under the action of a Lie/continuous symmetry group) seems to be inherited from the underlying projective geometry (although I could be wrong about that).

Dear Andrew Powell ,

You are right I may have been somewhat too quick (and not careful enough): The smooth structure cannot be characterized so easily. I had in mind spaces with symmetries which is a wider class: You replace then Lie-group by group.

But also smoothness has simpler characterizations than the full setting of differential geometry.

Dear Andrew Powell and readers,

Anyhow, to get back to the initial question:

There is a recurrent principle of category theory that appears in nearly every aspect it studies: The notion of monad. Each time you have something that resembles to Kuratowski closure operator there is a corresponding monad.

Zenon's paradox is based on either acceptance or non acceptance of such a closure principle.

Set theory is so universal because you make the strong assumption of completeness. Any union of sets is again a set. Union gives rise to a monad.

Now I wonder what the ontological status of such a closure operator is.

Can that which is infinite be included in something?

Dear Arno Gorgels

Finite set theory is also complete in this sense.

A finite union of finite sets is also finite!

Whereas to big categorical limits also might not exist for infinite sets.

Best,

TK

Dear Martin Klvana ,

Yes, consider the mapping n->n+1 from the natural numbers to the natural numbers.

Or the inclusion from the natural numbers into the set of reals.

Best,

TK

Thomas Krantz , but "the mapping n->n+1" makes only sense if n is a number, which infinity is not.

Dear Martin Klvana ,

I meant the mapping from the set of natural numbers to the set of natural numbers mapping n to n+1.

Otherwise concerning arithmetic with ordinals see

https://en.wikipedia.org/wiki/Ordinal_number

Best,

TK

Desr @Thomas Krantz

Can you say something more about monads in category theory and what their philosophical significance might be?

Many thanks.

Dear Andrew Powell ,

You may consult Mac Lane Categories for the working mathematician for the definition of a monad and the relation to adjunctions. This is very fundamental.

I write more in a forthcoming reply.

Best,

TK

Thomas Krantz , if ω is infinity, ω + 1 is absurdity. That which is infinite is immeasurable. Ordinal is a number, but infinity is not a number, hence infinity cannot be an ordinal number.