Hi. Let’s wait for more people to show up… Or it’s just us? OK. So listen up, I wrote this very cool post that gathers almost all my mathematical and philosophical interests in one place…

OK, whaterver, if you are not interested I will stop.

OK, let’s try again _ clears throat _ “The age-old mystery of Kant’s triads has baffled academics for centuries: what is the significance of the third element that finishes each of his triad. And is it possible that the categories were given to Kant by aliens?”

What, now it’s too dramatic? No, impossible, Kant never used any drugs! OK, whatever, I give up, no more introductions for these articles, let’s just get on with it!

The Mystery of Kant’s Triads

In The Critique of Pure Reason, Immanuel Kant introduces his famous list of pure concepts of understanding, also known as the categories, which, according to him, form the basic building blocks from which all other concepts are derived. These categories are divided into four groups: quantity, quality, relation, and modality. Each of these groups contains three categories, forming what Kant refers to as triads. For readers of Kant, the fact that each group contains three categories (instead of two) might seem strange, as Kant’s other classifications are typically based on dichotomies. What’s more perplexing is that the third category in each triad does not seem as obviously derived as the other two. For example, in the category of quality, the first two categories roughly correspond to “true” and “false” states, so what does the third one correspond to? According to Kant, it is a kind of blend of the two. Hmm…

For context, here is the list of Kant’s categories (with descriptions in my own words):

Quantity

• Unity (measure): Recognizing a thing as one
• Plurality (plurality): Recognizing there are several things
• Totality (whole): Unifying all things (of a given group) as one

Quality

• Reality: Recognizing presence
• Negation: Recognizing absence
• Limitation: Recognizing a thing as both present and absent (e.g., in different times)

Relation

• Inherence and subsistence: Representing a thing as being inherent or constant
• Causality and dependence: Representing causal effects (where things depend on each other)
• Community: Representing a reciprocal causality (where one substance causally determines another and vice versa)

Modality

• Possibility–Impossibility: Speculating whether a given thing can or cannot exist
• Existence–Non-existence: Perceiving whether something exists or does not
• Necessity–Contingency: Accepting a thing as inherently true (where existence is given through possibility itself)

Combination, But Not Entirely

Here’s what Kant says about these third categories:

“II. The number of the categories in each class is always the same, namely, three—a fact which also demands some consideration, because in all other cases division à priori through conceptions is necessarily dichotomy. It is to be added, that the third category in each triad always arises from the combination of the second with the first.”

But wait—if the third member of each triad is merely a combination of the other two, doesn’t that mean it’s not truly pure/atomic/first-order and should be excluded? Kant argues otherwise, although his explanation remains somewhat ambiguous:

Thus totality is nothing else but plurality contemplated as unity; limitation is merely reality conjoined with negation; community is the causality of a substance, reciprocally determining, and determined by other substances; and finally, necessity is nothing but existence, which is given through the possibility itself. Let it not be supposed, however, that the third category is merely a deduced, and not a primitive conception of the pure understanding. For the conjunction of the first and second, in order to produce the third conception, requires a particular function of the understanding, which is by no means identical with those which are exercised in the first and second. Thus, the conception of a number (which belongs to the category of totality) is not always possible, where the conceptions of multitude and unity exist (for example, in the representation of the infinite). Or, if I conjoin the conception of a cause with that of a substance, it does not follow that the conception of influence, that is, how one substance can be the cause of something in another substance, will be understood from that. Thus it is evident that a particular act of the understanding is here necessary; and so in the other instances.

What is this particular acts of the understanding for which Kant talks about and which enables the creation of each third category from the other two. For me it has to do with a peculiar process which converts empirical knowledge to universal knowledge, using the concept of the (understandable) universe (or “the world”) for which I want to talk about in this article. This process is based on faith, not knowledge, or more precisely, it is based on faith of knowledge.

If empirical knowledge involves constructing models, universal knowledge involves accepting those models as universally true. It is precisely this function that the third pure concepts of understanding enable us to accomplish.

Universal Knowledge and the Concept of the Universe

In this section, I aim to challenge Kant by proposing the following thesis: the third member of each triad of categories is not a pure concept of understanding. Rather, it serves as a bridge linking the other two concepts of understanding with reason. In other words, the third member enables us to form a mental image of the understandable universe (or substance, in philosophical terms), allowing us to apply the other two concepts to it.

To elucidate, consider the problem of induction (which I discuss in my text on Gettier and elsewhere), a problem that originally inspired Kant to write the Critique. This issue can be phrased as follows: We know our knowledge often becomes outdated and that what holds in one time/place/context may not apply in another. So, how can we justify the generalizations we make so frequently (which form the foundation of our thinking)? The simple answer: we cannot justify them. This has been known for millennia. The answer is so straightforward that it invites a follow-up question: How and why are we even able to make such claims in the first place? From where does our logical apparatus derive (an apparatus not shared by other animals), and what does it look like?

The answer lies in the concept of the understandable universe, or substance (I’ll use “universe” since the meaning of “substance” is unclear). The concept of the universe is both obvious and odd; after all, why do we have such a vivid notion of something we can never observe in its entirety?

The universe concept enables us to make general claims and to act as though we understand everything. General claims shape our understanding in a way that may be incorrect but remains useful; they allow us to make (sometimes incorrect but useful) predictions about our world.

Digging into Kant’s Epistemology

The section of the Critique where Kant puts forward his theory of knowledge, which occupies most of the book, is divided into two parts: transcendental analytics and transcendental dialectics. These parts correspond to the two main types of knowledge, according to Kant: understanding and reason. The main distinction between these is that concepts of understanding, based on categories, relate to possible experience, whereas concepts of reason, such as the soul, God, and the world, do not. For this reason, Kant sees concepts of reason as somewhat illusory.

A key question arises: to which of these two faculties does the concept of the universe belong? Kant explicitly classifies it under reason (if you accept that the concepts of “the universe” and “the world” are somewhat similar) but also implicitly places it under understanding (by including it in the categories). Personally, I view it as existing somewhere between the two. The concept of the universe is not a concept of experience because, although we experience the universe constantly, no one can claim to know something universally valid for all of it, neither practically nor theoretically. At the same time, the universe is not a concept of reason because it is a precondition for reason’s existence, just as causality is a precondition for experience.

But we are getting ahead of ourselves, so let’s start from the beginning - by tracing how the idea of the universe appears in each third category of Kant’s four triads.

Quantity

Unity

Plurality

Totality

Let’s start with the categories of quantity, or number. Categories originate from concepts in logic, and the categories in this triad are based on the logical predicates of singular, particular, and universal, often articulated as “one/unique,” “some,” and “all.”

The distinction between singular and particular predicates—and thus between unity and plurality—is clear. For example, it is the difference between perceiving 20 birds as 1 flock or as 20 birds. However, the distinction between these and totality is vast; they represent different types of knowledge. The first two types of knowledge have limited scope, while the third is universally valid.

To ellucidate this, let’s examine the predicates from classical Aristotelian syllogisms:

• Plurality - One A is B / Some A-s are B (limited scope)
• Totality - All A-s are B (universal scope)

This distinction becomes clearer: statements with limited scope can only be justified through observation. – if I observe any number of objects categorized as A (e.g., “apples”) and find they possess property B (e.g., “tasty”), I can conclude, based on those observations alone, that Some A-s are B or One A is B.

Statements with universal scope, on the other hand, are axiomatic by nature. Although All A-s are B and Some A-s are B differ significantly, the basis/reason for someone to concluding that All A-s are B are no different from the basis for saying Some A-s are B —

if I enjoy apples and have never had a bad one, I might say, “All apples are tasty,” whereas someone with the same experience but in a more skeptical mood might say, “All apples that I have eaten are tasty.” Our experiences are identical; my choice to assume universality is the only difference.

Thus, even both of them reference the same two variables (A and B), statements of the form Some A-s are B and All A-s are B are very different categorically. We might say that the latter type references a “secret third thing”—the universe. In other words, we could rephrase All A-s are B as All A-s in this universe are B.

This reveals a paradox: while the concept of the universe underlies empirical statements, it is not itself empirical, as we cannot make empirical observations about the universe (unless we are Laplace’s demon, but more on that later). And one way to navigate this paradox (actually, the only way I can think off) is to assume that universal statements are not entirely empirical—they create reality as much as they describe it. By saying all A-s are B-s, we are defining what A is, rather than simply describing the world.

For more details, see the second chapter of my time notes.

Quality

Reality

Negation

Limitation

The category of Limitation is often misunderstood because this triad is based on truth. Classical logic is bivalent: propositions are either true or false, yet this triad of quality has three categories. So, what is the third?

To approach this, let’s examine intuitionistic logic, where a proposition need not be strictly true or false. This connection is particularly interesting, as intuitionistic logic relates to category theory—a mathematical theory on which I wrote a book, also inspired by Kant’s categories (or perhaps Aristotle, it’s hard to say).

Anyway, classical logic, is rooted in Platonic epistemology which assumes that we humans have universal knowledge. Plato held that knowledge was given by God and that we recollect ideas already given to us. To postulate this universality, classical logic employs limitation, such as when everything is either B or non-B, meaning the B/non-B distinction is inherent to each object.

In contrast, intuitionistic logic represents the opposite view, that knowledge is subjective and limited in scope. Here, rather than contrasting reality with limitation, we use negation. A thing may be B or it may not be B, but it doesn’t have to be one or the other, because, since the predicate B is made up, there is no reason to think that it should apply to every object in the universe. This is a contrast to classical logic where all things are either B or non-b.

Again, in intuitionistic logic, when using reality and negation, I can make a statement that concerns a given object, such as saying it is B or not B. When using classical logic and limitation, however, I am making a statement about the entire universe, postulating it as divided into B’s and non-B’s.

This is how Kant puts it in “The transcendental clue to the discovery of all pure concepts of understanding”:

For example, if I say of the soul, “It is not mortal”—by this negative judgment I should at least ward off error. Now, by the proposition, “The soul is non-mortal,” I have, in respect of the logical form, really affirmed, inasmuch as I thereby place the soul in the unlimited sphere of immortal beings. Now, because of the whole sphere of possible existences, the mortal occupies one part, and the immortal the other, neither more nor less is affirmed by the proposition than that the soul is one among the infinite multitude of things which remain over, when I take away the whole mortal part.

Here, “the sphere of possible existences” essentially serves as a reference to the universe.

For further reading, see the chapter on logic in my Category Theory book.

Relation

Inherence

Causality

Community

We now turn to the dynamic categories, which (unlike quantity and quantity) concern not the phenomena themselves but the ways phenomena are perceived. In these triads, the relationship between the third categories and the rest differs slightly but still clarifies the idea of the universe.

The first triad is the category of relation, whose schema are the relationships between events in time.

The first category which (as all other first categories) is something like the base of the triad, is the category of inherence which determines the relations of appearances to time itself. The second category, the most famous one, is the relation of events based on cause and effect. And the third one concerns another relation - that of community or of interaction as Kant also calls it.

So, what is community? Let’s first see the explanation for the disjunctive logical statement on which it is based on, in “The transcendental clue…”:

Finally, the disjunctive judgment contains a relation of two or more propositions to each other—a relation not of consequence, but of logical opposition, in so far as the sphere of the one proposition excludes that of the other. But it contains at the same time a relation of community, in so far as all the propositions taken together fill up the sphere of the cognition.

Replacing “the sphere of the cognition” with “the universe” would bring us close to my initial thesis. The disjunctive relation here resembles limitation and the A versus non-A dichotomy we examined, though here we’re not discussing predicates, but relationships that change over time.

But there is something else we should note - this quote makes it seem like community, (which is based on disjunctions) is different from causality (which is based on consequence), but later Kant says that “nothing determines the position of anything else in time except that which is it’s cause?”

So, what is community? Later in the book, it is defined as reciprocal cause and effect: two substances exist simultaneously in community when each “contains within itself the causality of certain determinations in the other substance and, at the same time, the effects of the causality of that other substance.”

And where do the two approaches meet? Although the way Kant frames it is a little different, I view community as the idea of the universe as a system in which everything is in a causal relationship to everything else, including the things that are simultaneous (on which Kant stresses on the most.) It is the main “causal chain” which does not allow for the existence of other causal chain and demands that everything be connected to it in order to be considered real.

We don’t talk about this causal chain but intuitively we are always aware of it. For example, we see a phenomena that is at the same time unexplained and very real (it affects our senses very much), we often say that there must be some explanation for it. And I believe that the reason why we say that is because we are operating withing the bounds of the category of community.

Conclude with one more quote, this time from “Systematic representation of all synthetic principles of pure understanding”:

These are the three analogies of experience. They are nothing but the principles for determining the existence of appearances in time, according to all its three modes, namely:

• the relation to time itself, as a magnitude (magnitude of existence, that is, duration);
• the relation in time as a series (successively);
• and finally also the relation in time as a sum total of all existence (that is, as simultaneous).

Here again, we see the universe’s necessity, as time as a sum total requires a universal scope.

Modality

possibility/impossibility

existence/non-existence

necessity/contingency

The final triad, modality, is especially intriguing. Unlike the other three triads, it doesn’t deal directly with knowledge itself but rather with the relationship between knowledge and the knower—that is, you. Again, as in the other triads, the first two categories here create a clear dichotomy: possibility versus actuality. Possibility involves things that could theoretically happen, while actuality pertains to things that have indeed happened. This distinction mirrors Kant’s two types of logical propositions: problematic propositions (those without a necessary relation to reality) and assertoric propositions (those that you accept as true or false).

From these two types emerges a third one emerges: apodeictic propositions, or propositions that are necessarily true. In The Transcendental Clue, Kant explains this evolution:

“The apodeictical proposition cogitates the assertoric as determined by these very laws of the understanding, consequently as affirming à priori, and in this manner it expresses logical necessity. Now because all is here gradually incorporated with the understanding—inasmuch as in the first place we judge problematically; then accept assertorically our judgment as true; lastly, affirm it as inseparably united with the understanding, that is, as necessary and apodeictical.”

This passage outlines the three stages of evolution of our understanding, which eventually lead to the concept of the universe:

1. Initially, we formulate (make up) explanations based on what we observe and hear, without confronting these explanations with reality (we don’t expect things to make sense).
2. Next, we find that some explanations align well with reality, leading us to base some decisions on these explanations (we think it is possible for things to make sense).
3. Finally, we accept certain explanations as universally true, integrating them into our understanding. At this stage, we reject or ignore information that contradicts our established theories (we insist that things must necessarily make sense).

The transition from stages 1 and 2 to 3 represents a qualitative shift, similar to the difference between the categories of negation and limitation: possibility suggests that something may be true in some instances, while necessity implies it must be true in all cases.

Additional distinctions between stages 1,2 and 3 illustrate this evolution:

1. At stage 1 and 2, we operate on a phenomenological level, dealing only with things we directly perceive. In stage 3, however, we extend to imagining entities and concepts based on what we perceive but which cannot be directly observed.

2. At stage 1 and 2, our observations are fragmented. At stage 3, they coalesce into a cohesive whole, implying an underlying presumption that the universe itself forms a cohesive whole.

3. At stage 2, we can make random observations or discard observations as they lose relevance. At stage 3, however, every observation must connect with all the other ones (it is this connection which is necessary). New observations must align with our worldview, and we cannot disregard established observations without impacting our entire conceptual structure.

In essence, at stage 3, an observer feels they understand the fabric of reality, believing that, from some scattered observations, they can infer universal truths valid across all times and places. This assumption of universal knowledge is what Kant refers to in Systematic Representation…:

“That the connection of which with the actual is determined according to universal conditions of experience is (exists as) necessary.”

The concept of necessity embodies this belief: that the universe is comprehensible and that, furthermore, we have the capacity to comprehend it.

Where Universality Breaks

Consider this quote on negation and limitation:

“The whole sphere of possible existences, the mortal occupies one part, and the immortal the other.”

This sentence contains the assumption that all objects in the universe are either mortal or immortal, which is questionable e.g. what about the objects that are not alive in the first place? e.g. is a chair mortal or immortal?

This issue doesn’t arise in the dichotomy between reality and negation—for example, between “mortal” and “not mortal.” Here, it’s straightforward to categorize all possible objects: a chair, for instance, is clearly “not mortal” meaning that it doesn’t posess the quality of being mortal.

This is one of the many examples where the principle of universality breaks i.e. cases where our illusion that we possess universal knowledge leads us ashtray.

Universality and Bias

Whether we can truly understand without the categories of totality, limitation, community, and necessity depends on how we define understanding. But what we do know is that, even without these categories, we would still be able to accomplish quite a lot. We could still:

• Identify different phenomena (using the categories of unity/plurality),
• Classify some phenomena as present or absent in our perception (reality/negation), and
• Identify relationships among phenomena that allow us to see them as objects (inherence/causality).
• Finally, we could interpret what we observe and distinguish possible ideas from actual occurrences (possibility/existence).

Simple organisms are capable of surviving without concepts like totality, limitation, community, and necessity. And even for us, temporarily setting these categories aside can foster humility. You might even argue that removing them could make our thinking more objective, albeit less capable.

• Totality represents the bias toward universal claims that are valid for all objects in the universe, as opposed to more limited claims that concern our own environment. Without totality, we would remember that nothing we say is valid about all things or all people and at all times.

• Limitation represents the bias of applying systems of predicates to objects that don’t confront to this system (AKA boolean thinking). The loss of limitation, would prevent us from, seeing everything as being either or, according to some subjective criteria, when in reality many things can be both or neither.

• Community represent the bias of thinking that all events must be causally connected with all other events and excludes everything what doesn’t fit in this picture as unreal. So without it, we can have a richer understanding of the events around us.

• Necessity leads us to impose hierarchical relationships where none may exist. By abandoning the necessity for this hierarchy, we can put our preconceptions aside and see things more objectively.

So, if you take understanding to mean having an unbiased view of what we perceive, then one could argue that not only is understanding possible without these categories, but that these categories might actually cloud it.

Universality and entropy

So if those principles (which are based on universality) are wrong, then why have we adopted them? Well, as we said at the beginning, the existence of each third category follows from that of the other two. At first sight, that might look like a good-enough reason for them to “exist”. But, if you think about it, this line of thought is just an application to the same principle of universal knowledge that is embedded in all of those categories - considering something as necessarily true just because it follows from other things that we consider true i.e. disregarding the fact that we might be wrong and that every piece of knowledge has scope of validate (so it’s not universal.) So, just because we have the first category and we have the second one, does not mean that we have one that is a combination of the two. This is a problem that Kant himself acknowledges and talks about a lot in the transcendental dialectic - the problem that each piece of knowledge has a scope of validity and if you go outside of that scope, then the piece of knowledge becomes irrelevant.

But if our knowledge isn’t universal, how do we know where it ends?

That’s the root of all issues—we cannot know that.

Perhaps the only satisfactory (albeit wrong) answer to this question is that it somehow doesn’t end, and that we must be capable of somehow perceiving the whole world, the whole universe. This is the answer proposed by Plato, who posits a divine world of forms we can “recollect.” Kant, too, gives a similar answer by separating things as they are in themselves from things as they appear to us and talking about the second category. We can adopt this as an axiom of the world, which I’ll call the axiom of causality (I realize this name coincides with one of the categories, but I’ll use it here for its aptness). Accepting this axiom, the concept of the universe and each third category would logically follow from the other two categories and from this principle.

Alternatively, if we don’t insist on our answer being satisfactory (or want it to be 100% true), we could take the skeptic’s path and say that knowledge ends as soon as it begins—that we truly do not know anything. This perspective implies that everything we observe as a law is impermanent, and our knowledge would slowly disintegrate over time—a principle that I dub with the often-misused word entropy (which I am also probably misusing here.)

Laplace’s Demon and Its Archnemesis

The two different answers not only paint two different pictures of the universe that we are living in, but they describe two ways of seeing our role in this universe and our modus operandi. To end the article with a twist, I will write something about them here.

But first, let’s talk about entropy. We can say that the entropy of a given system corresponds to how easy it is to describe it’s state. e.g. an ordered deck of cards has a low entropy because it is easy to describe, while the only way to describe a shuffled one is to just make a list of all cards in the order in which they appear in the deck. The second law of thermodynamics follows directly from this definition - there are billions of ways a deck of cards can be unordered, but there are a just a few ways that it can be ordered. Thus, entropy increases with time (as we continue shuffling). The more we shuffle, the less accurate our description of the deck’s state becomes. A fascinating, often-overlooked fact is that this law holds true regardless of the method of description, or the type of order that the system has—whether precise (e.g., “the cards are in numerical order”) or imprecise (e.g., “the first half is red, the second half black”). Regardless of the simplicity or complexity of our description, it will gradually lose validity over time. This is entropy, and we observe similar phenomena as we move through space.

In one of his essays, philosopher and mathematician Pierre-Simon Laplace describes a godlike creature known as Laplace’s Demon:

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

If our knowledge were universal, humans might be considered “crippled” versions of Laplace’s Demon, theoretically capable of knowing and predicting everything but limited by perceptual and computational constraints. And (this is an extension of this idea that we see all the time) even if a person by themselves is not able to be Laplace’s demon, the humanity as a whole can collectively function as such. This thesis might sound weird, but it is the dream and promise of the human civilization and what many people are subconsciously subscribed to it.

But let’s now look into the unnamed archnemesis of Laplace’s demon that is hidden in a dark dual universe that is governed by entropy other than causality (I am not saying that it has to be called “Jencel’s Demon”, I am just putting the name out there and letting you make a decision). At first glance, this demon might seem inferior to the all-knowing demon of Laplace. Unlike Laplace’s demon, who seems to know everything, this demon does not comprehend any general principle for the way the universe works (except for the lack of such principles.) Unlike Laplace’s demon, which exist beyond time, this demon is always stuck in the present moment and it cannot really make any decision for the future, as it is not sure whether the knowledge it has in a given moment won’t become obsolete in the next one.

You might pity this incapable demon, but its situation is no worse than that of its powerful counterpart. Like Laplace’s Demon, it cannot be disappointed by anything, albeit for opposite reasons: Laplace’s Demon cannot be disappointed because it knows everything, while the other demon cannot be disappointed because it knows nothing. Entropy teaches this demon that all it knows and is will gradually vanish. In fact at every instant, this demon sees everything disappearing before its eyes.

Because this demon knows that nothing is universal, it feels no attachment to any principles it might discover, while Laplace’s Demon is intimately connected to the universe’s principles. Thus, for Laplace’s Demon, life is unchanging, while for the entropy-bound demon, it is ever-interesting.

You might consider this entropy demon an agent of chaos or a villain, as its personality mirrors that of many fictional villains. But it’s actually harmless, as it lacks the motivation to destroy anything or anyone. If reborn as a human, this demon would likely make an excellent Buddhist monk.

The only “evil” aspect, by modern standards, is its lack of inherent belief in progress or actions based on doctrines. But this view is not truly harmful. In fact, some of us might benefit from adopting it.

Not sure how to conclude, as there’s much more to say, but thank you for reading, and stay tuned for more updates.

Notes

• The Time Notes.
• Notes on Kant’s critique.