You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
ZXY (Mar 20 2026 at 05:54):Hello everyone,
I have a paper that touches the following foundational question: to what extent can distinctions between mathematical objects — for example between 0 and 1, or between different infinite cardinalities — be understood as purely structural rather than ontological?
From this perspective, questions related to the Continuum Hypothesis might be viewed as concerning the structure of mathematical frameworks rather than the existence of distinct kinds of infinity as such.
I would be very interested to know whether similar viewpoints have been developed within category-theoretic foundations, structural set theory, or related approaches. References would be greatly appreciated.
Thank you very much.
David Michael Roberts (Mar 20 2026 at 06:21):I note your preprint is titled "Triadic Definition and Explanation of Consciousness". I'm not sure if you intended to link to that, given your description talking about mathematical objects.
ZXY (Mar 20 2026 at 20:41):Thanks for the question.
The link is intentional — the paper is broader in scope, but it contains a specific ontological proposal about mathematical objects (numbers, infinity, etc.).
My question here concerns only that proposal, not the theory of consciousness as a whole.
More broadly, I am investigating whether philosophical analyses of consciousness can reveal structural assumptions underlying mathematics and mathematical objects.
ZXY (Mar 20 2026 at 20:47):For example, this passage in Section II：That 0 is 0, that 1 is 1, and that 0 differs from 1 are nothing but manipulations of M upon E. The so-called “infinity” is no exception. The differences between infinities spoken of by Georg Cantor—including the infinity of the number line itself—are nothing but differences between M and M, not between E and E. Therefore, the so-called Continuum Hypothesis (CH)—that is, whether there exists an intermediate infinity between different infinities—is nothing but a matter of the structure of M rather than of the ontology of E.
ZXY (Mar 21 2026 at 05:52):In other words, the real numbers are neither numbers nor points in a continuum. Ontologically, it is the E in itself.
Morgan Rogers (he/him) (Mar 23 2026 at 07:51):What do M and E stand for in the above?
ZXY (Mar 23 2026 at 10:46):M stands for Matter, E for Energy. Please refer to my paper for the definitions of matter and energy.
Morgan Rogers (he/him) (Mar 23 2026 at 11:24):Why does energy have an ontology while matter has structure? What does it mean to have "differences between Matter and Matter but not between Energy and Energy"?
Or more concisely, how do you delineate between energy and matter?
Morgan Rogers (he/him) (Mar 23 2026 at 11:35):The definition that you give of energy and matter in the paper is:
According to Ar. 1, “matter” can be defined as the O that is not the O of OC, denoted as M; and “energy” as the C that is not the C of OC, denoted as E.
Since this is not enlightening in itself, here is the referenced "Ar. 1":
In my view, life is neither a substance nor an attribute; birth and death are neither
modifications of a substance nor differences in an attribute. Life, as life, and birth and death,
as birth and death, can only be defined and interpreted as an irreducible change (Ar. 1):
• Life = OC （Ar. 1a）
• Birth and death ∈ ontology （Ar. 1b）
• Birth → death = ontological irreversibility （Ar. 1c）
In other words, birth → death is the irreversibility of being qua being. Without this
irreversibility, Ar. 0 is impossible.
Again, there is clearly some missing detail (saying "according to Ar. 1" earlier was misleading), so here is the definition of "OC" from even earlier on:
The ancient Greek philosophers articulated two kinds of irreducible change:
• a unidirectional and conserved change (denoted as C), as Heraclitus said: “No one can
step into the same river twice.” (DK B12)
• a reciprocal or symmetric change (denoted as O), as Plotinus said: “Being, in its very
activity, is not directed towards another, but returns to itself.” (Enneads VI.8.20)
Ontologically, besides O and C, there is a third irreducible kind of change—distinguished
from O by the asymmetry of change and from C by the directionality of change—which is
denoted as OC. (Zhang, 2020)
These are far from being crisp definitions which we can answer questions about.
Alonso Perez-Lona (Mar 23 2026 at 13:11):Morgan Rogers (he/him) said:
• a reciprocal or symmetric change (denoted as O), as Plotinus said: “Being, in its very
activity, is not directed towards another, but returns to itself.” (Enneads VI.8.20)
As a tangential note to OP, Plotinus is hardly an "ancient Greek philosopher". Neo-Platonism (or neo-platonisms) are, contrary to what the name would suggest, not a mere revival of Plato's ideas, far from that.
ZXY (Mar 23 2026 at 16:15):“The ontology of Category Theory” is a unified account that explains not only category theory itself, but also category theorists and the universe within which both emerge. My paper argues that three irreducible changes are sufficient for this unified account.
ZXY (Mar 23 2026 at 16:47):The three irreducible changes are called E (or energy), M (or matter), and OC (or life).
E = uncountable quantitative change.
M = qualitative change that introduces countable distinctions.
OC = irreversible direction in E and asymmetric transformation in M.
Morgan Rogers (he/him) (Mar 23 2026 at 17:21):ZXY said:
“The ontology of Category Theory” is a unified account that explains not only category theory itself, but also category theorists and the universe within which both emerge. My paper argues that three irreducible changes are sufficient for this unified account.
No it does not. The cited paper doesn't mention category theory at all.
ZXY (Mar 23 2026 at 21:17):E and M determine categorical structure, whereas OC determines the non-reversible, directional, and asymmetric aspects of morphisms.
ZXY (Mar 24 2026 at 05:47):To be is to become irreversibly. Can applied category theory model not only compositional structure but also the intrinsic irreversibility of real-world processes?
ZXY (Mar 24 2026 at 06:31):In other words, OC might be the bridge from structure to reality.
Morgan Rogers (he/him) (Mar 24 2026 at 08:50):To get a better idea of the content of what you're saying... what is an example of "O", an example of "C", and what is an example of "OC"? Are there examples that make clearer the distinctions you're trying to make?
ZXY (Mar 24 2026 at 16:42):Ontologically, E and M cannot be separated, but their difference can be illustrated by the contrast between continuous and discrete structures — for instance, between ℝ and ℤ — without identifying either with those mathematical objects.
ZXY (Mar 24 2026 at 16:57):OC confers irreversibility, directionality, and asymmetry on being, as seen in the irreversibility of time or entropy.
Morgan Rogers (he/him) (Mar 25 2026 at 08:29):That really doesn't answer my questions. Do the rational numbers "illustrate" E or M?
ZXY (Mar 25 2026 at 09:40):No. Rational numbers do not illustrate either E or M by themselves. They belong to M, not because they are “matter,” but because they are determinate structure.
More precisely: 1. E is not a class of numbers. It is what I call "conserved quantitative change". 2. M is not a class of physical objects. It is "discrete, determinate, symmetric structure". Rational numbers, like integers, are therefore better taken as "illustrations of M", because they are structured, distinguishable, and formally specifiable. 3. But no number, by itself, is E. At most, numbers can be used to model E.
So the point is not:
rational numbers = E
or
rational numbers = M
but rather:
rational numbers are a mathematical illustration of M-like determinacy,
If you want a very simple contrast:
M is what makes this number different from that number.
E is what makes quantitative variation possible at all without being exhausted by any particular determination.
That is why I use mathematics only as an analogy, not as an identification.
ZXY (Mar 25 2026 at 09:50):Category + historicity = reality