And again with this one. Maybe these types of diagrams are clearer with a few arrows. We’ll see! Please continue (re-)reading below:
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Category Archives: Philosophy
The Ancient and the Modern, V2
After creating the diagram for my last post, and appreciating the little arrows, I thought I would redo the diagram for this earlier post. No change in the text, so instead of duplicating the old post, I’ll just post the link to it.
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Jurgen Habermas
Further Reading:
https://www.npr.org/2026/03/14/nx-s1-5748043/jurgen-habermas-obituary-dead-96-german-philosopher
https://en.wikipedia.org/wiki/J%C3%BCrgen_Habermas
https://en.wikipedia.org/wiki/Communicative_action
https://en.wikipedia.org/wiki/The_Theory_of_Communicative_Action
Thanks for the inspiration: This image
[*2.41, *3.25, *3.26, *3.27, *5.54]
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The Yoneda Lemma
I’ve been wanting to post about the Yoneda Lemma of Category Theory for a while, and finally decided how to draw my little diagram. Next, I will sketch out the fundamentals of the lemma for you (not the proof though) even though there are many better presentations out there.
Then, oddly enough, I wondered what if anything the lemma could tell us about metaphysics. I know this is like using Godel’s Theorem to say things about epistemology (and so a category mistake, ha ha), but I still think it’s worth a thought or two. Apparently others have explored the same idea.
First, one would have to assume that reality can be modeled by some Category, so the Yoneda Lemma could apply to it. This may be in fact be wrong, but apparently Category Theory has been used to model various physical subsystems. If we just go ahead and assume it can, what could the Yoneda Lemma tell us?
- The Yoneda Lemma suggests an object’s essence (nature, being, identity) is entirely defined by its relationships (called morphisms or arrows in Category Theory) with all other objects in its system (i.e., its category), effectively equating an object with its web of connections. Metaphysically, this supports a structuralist or relational view where an entity has no intrinsic nature independent of its context.
- The lemma shows that knowing how an object relates to all other objects is equivalent to knowing the object itself. And so if two objects have identical relationships with all other objects, then they are functionally identical (isomorphic). This supports the idea that the existence of a thing is defined by structural behavior rather than intrinsic substance.
- The lemma works by embedding objects into relational mappings via their “view from elsewhere”—how they appear from the perspective of every other object. This could support a metaphysics where reality is fundamentally perspectival: objects are constituted by the totality of possible perspectives on them. There’s no “view from nowhere” that captures what something is independent of its potential relations.
The lemma was named after Nobuo Yoneda (1930-1996), a Japanese mathematician and computer scientist.
Further Reading:
https://en.wikipedia.org/wiki/Yoneda_lemma
https://ncatlab.org/nlab/show/Yoneda+lemma
https://blog.juliosong.com/linguistics/mathematics/category-theory-notes-14/
https://www.math3ma.com/blog/the-yoneda-lemma
https://curtjaimungal.substack.com/p/you-are-dual-to-everything-the-yoneda
https://temnoon.com/the-essence-of-relationality-yoneda-lemma-and-the-jeweled-net-of-indra/
https://en.wikipedia.org/wiki/Indra%27s_net
https://equivalentexchange.blog/2014/02/01/relations-all-the-way-down/
[*12.58, *14.160]
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Four Things to Avoid for Heijoshin
The four states of mind to avoid to maintain Heijoshin (平常心), or “calm, steady mind,” are called the Shikai (四戒), or “four sicknesses of the mind”. These are derived from Japanese martial arts and Buddhist philosophy and represent mental distractions that interfere with clear thinking and effective action.
The four things to avoid are:
- Kyo (驚) — Surprise: The mental disturbance that occurs when faced with the unexpected. It can lead to a moment of frozen indecision where one is unable to make correct judgments or take relevant measures.
- Ku (懼) — Fear: Being afraid of an opponent, an outcome, or a situation. Fear can cause your spirit to freeze and your body to lose the ability to move freely and effectively.
- Gi (疑) — Doubt: Questioning your abilities or judging your opponent’s actions, which leads to hesitation and indecisiveness. Doubt erodes confidence and delays decision-making.
- Waku (惑) — Confusion: A state of hesitation and bewilderment that prevents you from making quick, clear decisions and acting swiftly. This happens when the mind is clouded by a mix of indecision and doubt.
Further Reading:
[*14.134]
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Matter and Memory, Moment and Measure
The more of these moments memory can contract into one, the firmer is the hold which it gives to us on matter : so that the memory of a living being appears indeed to measure, above all, its powers of action upon things, and to be only the intellectual reverberation of this power. Let us start, then, from this energy, as from the true principle : let us suppose that the body is a centre of action, and only a centre of action. We must see what consequences thence result for perception, for memory, and for the relations between body and mind.
Perception is master of space in the exact measure in which action is master of time.
— Henri Bergson, from Matter and Memory
Further Reading:
https://brocku.ca/MeadProject/Bergson/Bergson_1911b/Bergson_1911_05.html
https://epochemagazine.org/50/creative-recollection-bergsons-theory-of-memory/
Click to download a pdf copy of Matter and Memory.
https://en.wikipedia.org/wiki/Matter_and_Memory
https://plato.stanford.edu/entries/bergson/#PercMemo
Emily Herring / Herald of a Restless World, how Henri Bergson brought philosophy to the people
https://share.google/aimode/qjOPizFa2d2U8Ab4v
https://www.ebsco.com/research-starters/literature-and-writing/matter-and-memory-henri-bergson
[*12.116]
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Aristotle’s Fourfold Division of Being
Aristotle’s fourfold division of being, found in his work Categories, classifies entities based on whether they are “Said of” a subject and whether they are “Present in” a subject. The four categories obtained by these distinctions are: Accidental Particulars (not Said of but Present in), Accidental Universals (Said of and Present in), Primary Substances (or Substantial Particulars) (neither Said of nor Present in), and Essential Universals (or Substantial Universals) (Said of but not Present in) (also called Secondary Substances).
| Present in | Not Present in | |
| Said of | Accidental Universals | Essential Universals |
| Not Said of | Accidental Particulars | Primary Substances |
Or, looking at these four categories another way, with explanation:
| Said of? | Present in? | ||
| Essential Universals | Yes | No | These are species or genera that are “Said of” particular substances but are not “Present in” any single subject. (i.e. the species of “humans”) |
| Accidental Particulars | No | Yes | These are non-substantial particulars that are “Present in” a subject but are not “Said of” any other subject. (i.e. the wisdom of Socrates) |
| Accidental Universals | Yes | Yes | These are universals that are both “Said of” and “Present in” subjects. (i.e. the wisdom that can be in some humans, such as philosophers) |
| Primary Substances | No | No | These are individual, particular things that can exist on their own, such as an individual man or other type of entity. (i.e. Socrates) |
Note the original term “accidental” can now be thought of as merely “non-substantial”. But Aristotle’s fourfold division wasn’t discussed at great length in the Categories, as pride of place was devoted to his ten-fold ontology, which we won’t be covering here.
The recent philosopher E. J. Lowe wrote much on essentially this same fourfold classification, which he called the Four Category Ontology. Lowe’s diagram of this, which he called “The Ontological Square”, also describes several other relationships between these four categories: Instantiation (between Kinds and Objects, and Attributes and Modes), Characterization (between Kinds and Attributes, and Objects and Modes), and Exemplification (between Attributes and Objects).
| Non-substances | Substances | |
| Universals | Non-substantial Universals, i.e. Attributes | Substantial Universals, i.e. Kinds |
| Particulars | Non-substantial Particulars, i.e. Modes | Substantial Particulars, i.e. Objects |
Further Reading:
https://philosophy-models.blog/2019/01/20/aristotles-categories-the-four-fold-division/
https://faculty.washington.edu/smcohen/320/cats320.htm
https://plato.stanford.edu/entries/aristotle-categories/#FouFolDiv
https://plato.stanford.edu/entries/aristotle-metaphysics/supp1.html
https://plato.stanford.edu/entries/aristotle-categories/
Ludger Jansen / Aristotle’s Categories
https://link.springer.com/article/10.1007/s11245-006-9009-1
Kevin Mulligan, Peter Simons, Barry Smith / Truth-makers in Philosophy and Phenomenological Research 44(3) 287-321 (1984)
E.J. Lowe / The Four-Category Ontology: A Metaphysical Foundation for Natural Science
[*4.94, *6.98, *6.143]
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Plato’s Fourfold Vision in the Phaedrus
In Plato’s Phaedrus, this “fourfold vision” refers to four distinct types of divine madness or inspiration; and being of divine origination are associated with various Greek gods. These are: prophecy from Apollon (later, Roman Apollo), mystic rites (telestic madness) from Dionysus (Bacchus), poetry and arts from the Muses (Camenæ), and love from Aphrodite and Eros (Venus and Cupid). These different forms of divine madness are seen as beneficial blessings and essential for a fulfilling life, crucial for living well and having a deeper connection to truth and beauty.
Also in the Phaedrus is Plato’s theory of the soul, most famously utilizing the metaphor of a chariot driven by two horses. The four types of divine madness represent different ways the soul can be uplifted. They are powers that can help guide the charioteer, steering the soul towards its higher purpose. If the soul follows the divine, the charioteer can ascend to the “plain of truth” (i.e. the realm of Plato’s Forms.) If not, the soul falls to earth in human form. The better the soul has seen truth, the higher will be its place in its “next” life.
It’s kind of odd that the charioteer is supposed to embody Reason, in order to steer the chariot’s horses properly, and yet there is no divine madness specifically associated with reason. Where is Athena (Roman Minerva), to aid the soul in its journey, as a guide for Reason? Perhaps Plato thought that Apollon could serve double duty, as he was thought to also symbolize reason as well as prophecy. Or perhaps the very notion of “divine madness” should not be associated with reasoning and logic. I disagree.
The writings of Benjamin Labatut come to mind as showcases of the “divine madness” of reason and logic. Various persons of science and mathematics are written about in a pseudo-bibliographical way: the mathematician Alexander Grothendieck, the computer scientist John von Neumann, and others. Other writers have also written about those that live in the highest realms of logic and mathematics: Kurt Godel, Grigory Perelman, etc. Their intellect may put them close to a form of madness.
Note:
I’ve chosen the Roman names for these Greek gods for the sole reason that they fit better in my diagram. I hope it’s not too confusing.
Further Reading:
https://en.wikipedia.org/wiki/Phaedrus_(dialogue)
https://en.wikipedia.org/wiki/Muses
https://en.wikipedia.org/wiki/Benjam%C3%ADn_Labatut
https://en.wikipedia.org/wiki/Alexander_Grothendieck
https://en.wikipedia.org/wiki/John_von_Neumann
Matthew Shelton / Divine Madness in Plato’s Phaedrus
https://www.academia.edu/118753337/Divine_Madness_in_Plato_s_Phaedrus
Katja Maria Vogt / Plato on Madness and the Good Life
Click to access paper-vogt-plato-on-madness-and-the-good-life.pdf
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Plato’s Fourfold Division in the Philebus
In Plato’s dialogue the Philebus, Socrates uses a fourfold division to describe fundamental, metaphysical principles of all things: the Unlimited (ápeiron, the indefinite, boundless aspect), the Limited (péras, the definite aspect of measure and boundary), the Mixture of these two (meikton, a combined indefinite-definite), and the Cause of the mixture (aitia, which gives structure to the mixture and is responsible for unifying it).
This framework provides the foundation for the dialogue’s central argument about what constitutes the good life, and what activity is best to achieve it. In the dialogue, Socrates uses the four parts of the division to examine the natures of both pleasure and reason, determining that their mixture, brought about by some cause, is the key to a truly good life. So neither pleasure alone, nor reason alone, would suffice.
This fourfold division reminds me of Aristotle’s later schema of the Four Causes. Initially I thought that the Unbounded could be the Material Cause, the Bounded could be Formal Cause, the Mixture could be Final Cause, and finally the Cause could be Efficient Cause. That is because one might think that there is always more material to be had if you want it, and the form of a project puts shape, constraints, limits, and relationships on it.
However, I now believe that what is truly limited is the raw material (matter, energy, time, information), and what is unlimited is all possible form (and not any one particular form). The combinations or permutations that one can make from a finite set of elements are essentially unlimited in number, or at least increase exponentially with the length of the permutation or the size of the combination.
Consider the finite nature of an alphabet, and the endless number of sentences that can be formed from it. Or think about the four DNA base pairs, and the endless variety of genes and organisms that can be generated from them. Remember Darwin’s quote: “Endless forms most beautiful”. Of course many of sentences one can make are gibberish, and many of the organisms one could make are not fit or even viable.
I realize that in Plato’s dialogue the Unlimited and the Limited are more about the continuous and the discrete: the infinite and the finite, the unbounded and the bounded, the indefinite and the definite. The examples of the Unlimited given have the common aspect of “more and less”, i.e. they are indeterminant. By contrast, the examples of the Limit(ed) are of a determinant nature, a bound, a limit, a fixation, a stop to the “more and less”.
It seems that one could spent an unlimited amount of time studying the intricacies of the Philebus and thinking about the meaning of it. But I am starting to feel that my time is indeed limited and other matters require my attention. I know I am probably at odds with all others in the proper interpretation of this part of the dialogue. At least you might enjoy the limited links I have provided, and you still have the unlimited freedom to make of them what you will.
Further Reading:
https://en.wikipedia.org/wiki/Philebus
https://www.gutenberg.org/files/1744/1744-h/1744-h.htm
https://standardebooks.org/ebooks/plato/dialogues/benjamin-jowett/text/philebus
https://plato.stanford.edu/entries/plato-ethics/
Mary Louise Gill / The Fourfold Division of Beings (Philebus 23b-27c)
https://www.academia.edu/40402669/The_Fourfold_Division_of_Beings_Gill_on_Platos_Philebus_
Ashley Lascano / A critical analysis of the metaphysics of limit and unlimited in Plato’s Philebus
https://commons.lib.jmu.edu/honors201019/691/
George Rudebusch / Philebus 23c-26d: Peras, Apeiron, and Meikton as Measure Theory
https://impactum-journals.uc.pt/platojournal/article/view/10332
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The Golden Mean
The Golden Mean is the desirable middle ground between two extremes of behavior, one of excess and the other of deficiency. It appeared in Greece at least as early as the Delphic maxim “nothing in excess”, which was discussed by Plato. Aristotle analyzed the Golden Mean by saying that the virtues of character can be described as means between having too much and having too little. For example, courage is a virtue in the Aristotelian view, being the mean between recklessness (too much) and cowardice (too little).
- Excess
- Dearth
- Virtue
- Mean
The Mean between Excess and Dearth is Virtue!
Further Reading:
https://en.wikipedia.org/wiki/Golden_mean_(philosophy)
https://philosophybreak.com/articles/the-golden-mean-aristotle-guide-to-living-excellently/
https://plato.stanford.edu/entries/aristotle-ethics/
[*14.98]
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Equivalent Exchange 
