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: Metaphysics
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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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 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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Four-dimensionalism
Previous posts (here and here) have considered mathematical and scientific ideas about four dimensions. Here are some links about the metaphysical considerations of four-dimensionalism.
Further Reading:
Theodore Sider / Four-Dimensionalism: An Ontology of Persistence and Time
https://ndpr.nd.edu/reviews/four-dimensionalism-an-ontology-of-persistence-and-time/
Ludwig Jaskolla / Real Fourdimensionalism: An Essay in the Ontology of Persistence and Mind
https://en.wikipedia.org/wiki/Four-dimensionalism
https://plato.stanford.edu/entries/temporal-parts/
https://en.wikipedia.org/wiki/The_Unreality_of_Time
https://en.wikipedia.org/wiki/Philosophy_of_space_and_time
Also:
https://ndpr.nd.edu/reviews/persistence-through-time-and-across-possible-worlds/
https://ndpr.nd.edu/reviews/how-things-persist/
And even:
https://ndpr.nd.edu/reviews/the-language-and-reality-of-time/
https://ndpr.nd.edu/reviews/the-elements-and-patterns-of-being-essays-in-metaphysics/
https://ndpr.nd.edu/reviews/a-materialist-metaphysics-of-the-human-person/
Maybe:
https://www.friesian.com/lieb.htm
[*10.89]
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Schopenhauer’s Four Laws of Thought
The first three of Arthur Schopenhauer’s Four Laws of Thought are pretty much the same as the classical three laws of thought. Schopenhauer added a fourth law that was basically for his Principle of Sufficient Reason.
- Identity
- Non-contradiction
- Excluded middle
- Sufficient reason
These Four Laws are often given in two flavors: the first, in fairly concrete terms of subjects and predicates, and the second, more glib in terms of existence and being and such (isness).
- A subject is equal to the sum of its predicates. Everything that is, exists. (Identity)
- No predicate can be simultaneously attributed and denied to a subject. Nothing can simultaneously be and not be. (Non-contradiction)
- Of every two contradictorily opposite predicates one must belong to every subject. Each and every thing either is or is not. (Excluded middle)
- Truth is the reference of a judgment to something outside it as its sufficient reason or ground. Of everything that is, it can be found why it is. (Sufficient reason)
The phrase ‘it can be found’ sounds like a constructive method rather than a mere existence proof, but the common theological technique that combines both by saying “everything happens for a reason” avers the reason to an ineffable deity. (I bet Schopenhauer would have disliked this view because from what I understand he was an atheist.)
Moving on, I would like to represent these four laws in even more concrete terms of logical expressions. In the following attempt, let a, b be subjects (or objects), and P, Q be predicates (or qualities):
- ∀a (a ≡ ∀P P(a))
- ∀a ¬∃P (P(a) ∧ ¬P(a))
- ∀a ∀P (P(a) ∨ ¬P(a))
- ∀a ∃b (b → a)
When detailed in this way, these four laws don’t seem very complete, or don’t quite form a unity, as implication and equivalence are each in only one of them. Even though it doesn’t help that criticism, perhaps one can succinctly say:
- Things can be reduced to (all) their qualities.
- Qualities are disjoint from their opposites.
- Qualities and their opposites are sufficient.
- Things are entailed by some thing (possibly same).
In addition, I quite liked this Goodread review which aligns Aristotle’s Four Causes with Schopenhauer’s Fourfold Root. So then:
- From Parts : Material Cause : Becoming : Identity
- For Functions : Final Cause : Knowing : Non-contradiction
- Into Structures : Formal Cause : Being : Excluded-middle
- By Actions : Efficient Cause : Acting : Sufficient reason
Further Reading:
https://en.wikipedia.org/wiki/Law_of_thought
https://en.wikipedia.org/wiki/Law_of_identity
https://en.wikipedia.org/wiki/Law_of_excluded_middle
https://en.wikipedia.org/wiki/Law_of_noncontradiction
https://en.wikipedia.org/wiki/Principle_of_sufficient_reason
https://plato.stanford.edu/entries/sufficient-reason/
https://plato.stanford.edu/entries/schopenhauer/
https://en.wikipedia.org/wiki/Quantifier_(logic)
https://en.wikipedia.org/wiki/Universal_quantification
https://en.wikipedia.org/wiki/Existential_quantification
https://www.goodreads.com/review/show/1328220242
Schopenhauer’s Fourfold Root of the Principle of Sufficient Reason
[*11.196, *11.197]
Notes:
At some point, I need to understand the difference between the law of the excluded middle and the principle of bivalence.
https://en.wikipedia.org/wiki/Principle_of_bivalence
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Everything is Four
Is everything four? Some try to prove it with a numerological trick. Take a word. Count its letters. Convert the number to words. Count letters. Repeat. Every English word seems to end up on four or 4, with nowhere else to go! Voila!
Since I’ve searched for this topic, a musician has released an album with this title. Cool!
But what do I mean by it? Everything can be divided into four parts, or has four aspects, or four sides, or what? I’m not sure, exactly.
But let’s test it against Alfred Whitehead’s Criteria for Metaphysical Theories!
- Is it consistent? Yes! That is, nothing in the theory contradicts other parts of the theory, because there are no other parts. And if something is part of a foursome, that something can also be a foursome (even if an arbitrary one).
- Is it coherent? Yes! That is, the theory is logically whole, such as it is. A bit boring? Perhaps…
- Is it applicable? Yes! That is, we can apply our method to reduce something to four parts to everything, as long as we don’t care what the parts are. Plus we can combine anything with three other things, ad nauseum!
- Is it adequate? No, not really. It does little to explain itself or the rest of the world.
So, we must continue our search for our ultimate metaphysical theory. It must be everything is four, plus something else… plus two more somethings…
Further Reading:
http://www.unterzuber.com/4our.html
http://www.marijn.org/everything-is-4/
https://en.wikipedia.org/wiki/Everything_Is_4
Whitehead’s Criteria for Metaphysical Theories
[*11.140]
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Writing the Book of the World
If one was writing a book that described the entire world or universe as it is, how should that book present the world to us? It is not enough to speak truly, itemizing “all that is the case”, one must also use the right notions while doing so.
Philosopher Theodore Sider wants us to accept structure as the all important fundamental notion of how to talk about the world. His idea of structure is that it reveals where the joints or articulations of the world can be carved, and that the structure of the world is real and it is objective. Structure is the right and proper way to find these joints, and go about this carving.
(Of course the structure that Sider promotes is not to be confused with the structuralism of linguistics and anthropology that was so popular before deconstruction and post-modernism critiqued it nearly to death. This post is not about structuralism because its structure is a reflection of language and the mind itself, not an attribute of the actual world.)
Chapters 1-8 are titled: Structure, Primitivism, Connections, Substantivity, Metametaphysics, Beyond the Predicate, Questions, and Rivals. Chapters 9-12 are devoted to ontology, logic, time, and modality (because I guess these are favorite topics in metaphysics) and what structure tells us about them. I don’t think Sider is saying that reality is carved naturally into these four domains, but I think it makes a rather nice fourfold.
In Metaphysics, Ontology is another word for Being, but it can also mean a classification system for the different kinds of things that exist (but I guess that’s not metaphysical). Modality is the Metaphysical or Epistemic study of necessity and possibility, so it is certainly related to time. There are also modal logics which have quantifiers for modalities such as necessity and possibility.
I closing I must say that once you carve up some structure by its joints, then you are left with parts, which may be structures in their own right. And as I’ve posited elsewhere, functions and actions are the structures and parts of time. I also wonder if there is a comparison of Sider’s structure to the metaphysics of E. J. Lowe, but perhaps I should just read each of their work.
Further Reading:
Theodore Sider / Writing the Book of the World
Some reviews:
http://ndpr.nd.edu/news/writing-the-book-of-the-world/
Click to access 12-05-wtbotw-review.pdf
https://en.wikipedia.org/wiki/Ontology
https://plato.stanford.edu/entries/modality-varieties/
https://plato.stanford.edu/entries/modality-epistemology/
https://en.wikipedia.org/wiki/Structuralism
Review that needs registration to read:
https://www.lrb.co.uk/v34/n16/aw-moore/carving-at-the-joints
[*9.8, *11.112]
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Noether-Pauli-Jung, V2
What happens when the fourfold of Noether’s Theorem is spliced together with the fourfold of Pauli-Jung? Both have Space-Time and Matter-Energy. The former has Conservation and Symmetry, and the latter has Causality and Synchronicity. And if Space-Time and Matter-Energy are both divided into Space and Time and Matter and Energy, one obtains this eight-fold.
Causality means that some action or cause in time (say a process) of things in space can have an effect (another process, say) on different things in space, and Synchronicity means that different events (say processes) separated in space can have non-causal relationships between them. Conservation means the consistency of a quantity of matter or energy or matter-energy through time, and Symmetry means the consistency of a measure of a structure or form through space.
I am reminded of my fourfold Four Bindings, consisting of Chains, Grids, Blocks, and Cycles. Causality and Synchronicity are Chains (or non-chains for the latter) Space and Time are Grids (or flexible meshes), Matter and Energy are Blocks (or chunks of stuff), and Symmetry or Conservation are Cycles (of the group-theoretic kind or the equivalence class kind or just loops).
Further Reading:
https://en.wikipedia.org/wiki/Causality
https://en.wikipedia.org/wiki/Synchronicity
https://en.wikipedia.org/wiki/Symmetry
https://equivalentexchange.wordpress.com/2012/05/04/noethers-theorem/
https://equivalentexchange.wordpress.com/2018/01/23/atom-and-archetype/
This is a reworking of a previous six-fold diagram that I believe is served better as an eight-fold.
[*10.68, *10.155, *11.55]
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Equivalent Exchange 