In relation to the thesis of this book, bottom-up processing is encapsulated by the mathematics of set theory, and top-down is encapsulated by mereology. The first is the science of making sets from elements, and the later is the science of making parts from wholes. In both cases, sets and parts are understood in terms of some mental or physical space (that space is an essential and a priori feature of mind has been argued by scholars such as Longchenpa and Kant).

From a mathematical perspective, either bottom-up or top-down would be sufficient, since either framework (mereology or set theory) is provably complete in a mathematical sense. In fact, mereology and set theory were formulated independently at about the same time in history, and set theory “survived” in mathematics because it was felt that only one of the two was necessary. From a psychological perspective, however, it is clear that cognition relies heavily on both bottom-up and top-down processing. Therefore, if we wish to bring mathematics in line with cognitive science, we must give an account of the top-down process.

Cognitive Set Theory is a first attempt to do so. In analogy to the point, the starting point of point-set topology, we have posited “everything”, the building block of mereotopology. In that attempt, bottom-up processes were assumed to operate “prior” to top-down processes, but that is probably disputable. The significant distinction seems to be that the top-down process is conceptual, and as such, is better modeled by a non-transitive structure such as sets (whereas the parallel-processing bottom-up structure requires a transitive structure: parts).

In a nutshell, that is how bottom-up and top-down processing might be modeled as mathematical spaces. Excuse the lack of formality of this essay, but I have been intrigued by recent work in mathematics and psychology, and I believe it is important to clarify from the start of such an endeavor how mathematics is both suitable and unsuitable in its current formulation.

]]>In the philosophy of mathematics, the abstraction of

actual infinityinvolves the acceptance (if the axiom of infinity is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted withpotential infinity, in which a non-terminating process (such as “add 1 to the previous number”) produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps.

It is important to note that both systems are happy to use the term infinity: if they did not believe in infinity, then they would hold a different mathematical position called finitism. They use the term in significantly different ways, however. Mathematicians who believe in completed infinities (like Aleph Null) treat infinity like a number. Mathematicians who do not accept completed infinity, and believe only in potential infinity, treat infinity like a limit process (which has a more complicated representation than a finite number).

The subject of this article is a paradoxical statement which is widely held to be true by mathematicians who believe in a completed infinity:

1) $latex 1.0 = 0.99\overline{9} &s=1$

Most people are not intuitively comfortable with this statement, because it implies that two different numerical (decimal) representations correspond to a single number. The “proof” looks as follows:

2) $latex x = 0.99\overline{9} &s=1$

3) Multiply both side by 10 to get : $latex 10x = 9.9\overline{9} &s=1$

4) Subtract (3) from (2) to get: $latex 10x-x = 9x = 9.9\overline{9} – 0.99\overline{9} = 9 &s=1$

5) Therefore, $latex x = 1 &s=1$

There are at least two ways out of this paradox. Most mathematician these days believe in completed infinity, so they are forced to accept (2) & (5) and therefore (1). If you hold an intuitionistic view of mathematics, however, you can get rid of the paradox by handling infinite representations as limit series.

Before we go farther, I should mention that whether or not my reasoning is sound, it is mathematically novel (at least according to some of my mathematician friends). If you happen to have conversations about intuitionism with narrow-minded mathematicians, they will not be sympathetic to your view (despite the fact that a number of genius mathematicians have advocated this stance, such as Arend Heyting, L. E. J. Brouwer, Stephen Kleene, and Henri Poincare).

To return to how an intuitionist might represent a number which has “infinite length”. Obviously, a representation like $latex 0.99\overline{9} &s=1$ where the nines run “to infinity” will not be acceptable. However, we can turn it into a limit series as follows:

5) $latex \sum\limits_{n=1}^{\infty} \frac{9}{10^n} &s=1$

In other words, repeating decimal representations are written as a limit where the number of terms *approaches* infinity. The significant difference between the limit representation and the repeating decimal representation is that the former is more clearly seen to be a process, not a “number whose length has a cardinality of aleph null”, which again is disallowed (in intuitionism). Given this representation, let’s re-write (4):

6) Subtract (3) from (2) to get: $latex 10x-x = 9x = 10 * \sum\limits_{n=1}^{\infty} \frac{9}{10^n} – \sum\limits_{n=1}^{\infty} \frac{9}{10^n} = \sum\limits_{n=1}^{\infty} {\frac{9*10}{10^n} – \frac{9}{10^n}} = \sum\limits_{n=1}^{\infty} \frac{81}{10^n} &s=1$

The result from (6) is not 9, it is (commonly written as) $latex 8.9\overline{9} &s=1$, and for an intuitionist, it is 8.1 after one term, approaching a limit of 9 only “at” infinity (which, on that account, we never reach, and therefore we do not find that (1)). Now, why did we arrive at a different result when we used a limit formula, instead of the subtraction formula? My “intuitive” answer is that we paid attention to the number of terms in the series expansion. In other words, when in (4) we perform the subtraction $latex 9.9\overline{9} – 0.99\overline{9} &s=1$, we are subtracting term *i* in the second series expansion from term *i+1* in the first series expansion (i.e. we are ignoring the first term). While this might be OK if you have a completed infinity of number of terms, an intuitionist account must yield a correct series at every (finite) step, and combine terms under the appropriate loop variable, *i*. That means that after three terms, we have:

7) $latex 9.99 – 0.999 = 9 – 0.009 = 8.991 &s=1$

In other words, $latex 9x \neq 9 &s=1$, and as a result, intuitionists who do not believe in a completed infinity do not have to accept that $latex 1.0 = 0.\overline{9} &s=1$.

OK, mathematicians who can either logically tear me apart or pat me on the back, I welcome your feedback.

]]>The slides for the summary are probably the best starting point:

The summary covers a slightly narrower scope with more text:

]]>So, I have written a short book about a cup, called “Cup: Reflected”. It’s about 30 sentences, and has 10 pictures. It serves as an introduction to the concepts of “perceptual references” and “conceptual references” through the use of mirrors and maps.

Take a look at the 4×6 PDF here: https://cognitivesettheory.com/files/CupReflected.pdf

]]>Boolean logic is a fantastic symbolic calculus, and we will not challenge its basic tenets. However, it is not sufficient to adequately describe all of our cognitive operations: the human mind is composed of more than just the rational mind.

In particular, Boolean logic applies to singular (unitary) subjects, and from a Madhyamaka point of view, there are no non-composite subjects in reality. Although the conceptual mind imputes singularity, all existing objects are composed of a plurality of parts. Boolean logic, for all of its merit, was not designed to apply to non-singular subjects.

So from the Buddhist point of view, modern logic is not incorrect, but it is a logic which applies only to concepts. A more general logical system with four outcomes, called the catuskoti (meaning “four corners”) has been used for debate in India before Buddhism began.

The catuskoti offers a plurality of logical outcomes: {true}, {false}, {true,false} and {} (i.e. neither true nor false). Some things are also said to be beyond this analysis (ineffable), but we will of course not speak of those. A modern logician would of course balk at the notion of “true and false” and “neither true nor false” as logical outcomes, since they are not logical outcomes with a Boolean algebra: they are contradictory within a logical system in which the premise must resolve to a single truth value.

Boolean algebra is appropriate to synthetic (a priori) statements or deductions. To use a classical example, if all men are mortal, and Socrates is a man, then we may conclude with certainty that Socrates is mortal. Statements about the world, however, are less certain. For example, the way that the name of an object corresponds to its object poses a problem known in philosophical circles as the problem of identity. To put it simply, if a name is taken to refer to a particular thing, and that thing changes while the name does not, then we have some uncertainty with respect to the correspondence of the name and the object.

Let us take an egg as an our subject of analysis. Conceptuality, the egg is a single thing, but of course it corresponds to a region of space filled with egg-matter which is multiple (or at least divisible). We then apply the logical predicate “yellow” to the subject, which corresponds to the question: is the egg yellow? On a classical Boolean analysis, we have the following possible answers:

- True: the inside of the egg (the yolk) is yellow.
- False: the outside of the egg (the egg white or shell) is not yellow.

If we are determined to reduce this answer to a single truth value, we might say that these two answers should be combined with one of the Boolean operators, either AND or OR:

- True: the egg is yellow because some part of it is yellow (true OR false).
- False: the egg is not yellow because some part of it is not yellow (true AND false).

There is a problem with this logic, however. Suppose we have another predicate, not-yellow, and ask the same question:

- True: the egg is not-yellow because some part of it is not-yellow (true OR false).
- False: the egg is not not-yellow because some part of it is not not-yellow (true AND false).

The problem should be clear: we obtain the same answer asking if the egg is yellow that we do when we ask if it is not-yellow, which entails a contradiction (yellow, because it is a non-compound predicate, is subject to the law of the excluded middle). There is simply no way to simplify the situation (consistently and nonparadoxically) when we start with a subject that is non-uniform with respect to the answer.

A mereological logic (which is better suited to traditional Buddhist analysis) a would answer the question in terms of parts of the object, in which case we have:

- {True, False}: the egg is yellow because a part of it is yellow, and it is not yellow because a part of it is not yellow.

Again, the answer {true,false} is the result of applying a binary (logical) predicate to a *compound* subject: an object which has the parts {yellow, not-yellow}. In traditional Buddhist texts, this answer is expressed as the combination of a predicate and its negation, or “yellow and not yellow”.

There is one more outcome to the catuskoti argument, which is “neither yellow nor not-yellow”. One example of a subject which results in this answer is a word that does not correspond to any actual object. Take, for example, the horns of a rabbit:

- The horns of a rabbit are not yellow
- The horns of a rabbit are yellow

Both of these statements are equally meaningless: a rabbit does not have horns. Therefore, there is no basis to say anything about them: since they do not have a color whatsoever, we cannot say that this non-existant color is yellow or not yellow.

When something and it’s nonexistence

Both are absent from before the mind,

No other option does the latter have:

It comes to perfect rest, from concepts free.

~Shantideva

**Bibliography**:

- Graham Priest, Beyond True and False
- Shantideva, Way of the Bodhisattva

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There are two pages on this blog with related content:

There is also a post about the tetralemma (or catuskoti), and how that might be brought into accord with modern logic (or vice versa):

]]>In the Western philosophical tradition, this debate has focused on the reality of particulars (or concrete things) vs universals (or abstract things). In early theories of both Buddhism and Mathematics, spatial regions are identified by the points (or “partless particles”) they contain, where those points are taken as primitive (either as axiomatic within geometry, or as the atoms of physical reality). In other words, particulars (as opposed to abstract things) are real, and the smaller the better.

Later Buddhist theories arrive at the conclusion that atoms are no more real than larger objects, in that both are imputed (or nominal) entities. Several recent Mathematical theories similarly abandon points in favor of point-free topologies. So both ontologically and mathematically, there has been movement away from their respective reductionist positions (which define objects in terms of their parts, and therefore ontologically depend of partless entities). Since Cognitive Set Theory similarly adopts a more holistic (instead of reductionistic) view, we present a few more details about the evolution of Buddhist psychology, and how that might map onto mathematics.

A simplified version of the evolution of the theory of svalaksana proceeds as follows:

- According to Pramana theorists, the real is simple, or singular (“eka”): we might say more precisely that what is real is spatiotemporally contiguous: a thing which is whole is not distributed. Mathematically, we will call it a well-connected region in N-space.
- According to the Sautrantika view (or the “external realism” of Dunne), there are “real” external things: momentary, partless particles. This view has been expounded in depth by Tscherbatsky [Buddhist Logic Vols 1&2], and it is compatible with the mathematical theory of points (which cannot be divided).
- According to Chittamatra and Madhyamaka views, the objects of awareness are of the nature of mind, and only imputed. In other words, there is no
*a priori*reason to divide reality into a certain set of objects as opposed to another. In other words, space is no more composed of points than it is composed of bricks: tiny and large particles are equally real (or equally unreal).

To establish the spatial analogue to the later Buddhist schools, we summarize the divergence from point-space as follows:

- Regions are well-connected
- Space is not composed of points
- Regions are imputed

The first and second tenets are easily established in mereotopology, where space is divided into regions that are not (inherently) composed of points.

We alter the third tenet by saying that an object only exists in virtue of an imputed boundary. Since that boundary does not exist as an object within the original space, we say that it is an imputed boundary. The details of how an object (the boundary) may demarcate events within a space without being an actual object within that space is taken up in more detail in another blog post.

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- Space is represented as an open-dimensional, or multi-dimensional, manifold.
- Objects are demarcated by boundaries. A boundary is an imputed object which has zero width (it is a thing of dimensionality N-1 in an N-dimensional space).
- The division of space into objects is
*not*set-like: it has the relationship of parts to wholes, and is transitive. The complete definition of an object entails both its larger context and its boundary (a separating hyperplane or hyperspace). - The boundary of an object may be characterized by a function. That function is not necessarily characterized intrinsically (i.e. by the contents of that object), since it can also be specified extrinsically.
- Objects, whether they are regarded as real or merely perceptual, serve as the elements of sets (or the ultimate contents of concepts).
- Sets may be labeled, and collected into higher-rank sets.

To understand Cognitve Set Theory in Buddhist terms, the part and chapter titles can be reformulated as follows:

- Dharmas (Alaya, Dharmas, Shunyata)
- Kayas (Dharmakaya, Sambhogakaya, Nirmanakaya)
- Mirrors (Percepts of Objects, Concepts of Percepts, and Concepts of Concepts)

The translation of “everything” as Alaya is straightforward: Alaya is often translated as “ground” or “basis of all”). For the schools of Buddhism before chittamatra, “everything” might also be translated as svabhavikakaya. “Dharmas” literally means “things”, and “shunyata” literally means “nothingness”, so those translations are quite close. The translation of the Mirrors section is more or less literal. The translation of the physical, perceptual, and conceptual universes as the Kayas, however, requires some justification.

In Buddhism, the three spheres in which a Buddha manifests are the Dharmakaya (“truth body”), the Sambhogakaya (“bliss body”), and the Nirmanakaya (“emanation body”). These are held to be decreasingly pure and only seen by Buddhas, such that most beings only see the least subtle level (the nirmanakaya).

The Dharmakaya is that Kaya which contains all of the others, and it is sometimes translated as the ground (or basis) of existence. Since this ground is non-dual, there is a sense in which it does not matter if we call it mind or matter (as long as we do not confuse *multiple* non-dual doctrines which have adapted different terminology). Followers of the chittamatra school refer to it as mind, whereas materialists would refer to it as matter: in both cases, this non-dual nature must have the qualities of both matter and mind.

Given that Dharmakaya is referred to as mind, it is no surprise that Mahayana Buddhists group the other two Kayas into the Rupakaya (which means “form body”, or perhaps “material body”). The Sambhogakaya eminates from the Dharmakaya. It’s literal translation is the “bliss body”, and is related to the sensation that a Buddha or Bodhisattve experiences in virtue of their purity. Sensation is held by Buddhist thinkers to be a nonconceptual consciousness which anticedes the (potentially) conceptual consciousnesses that follow it. Finally, the Nirmanakaya eminates from the Sambhogakaya. It is the least subtle body, in which Buddhas manifest in order to teach beings who are are not pure enough to perceive the other realms.

Since the descriptions of the Kayas can be quite nebulous, and I am not realized enough to know them directly, these chapter headings can also be defined in terms of the four foundations of mindfulness, or the eight consciousness model. In terms of the four foundations of mindfulness, the chapters correspond to the first three: body, sensation, and mind. Using the eight consciousness model, the physical universe corresponds to the realm of the sense objects (which, according to the Sautrantika view, is hidden from direct perception). The perceptual universe corresponds to the five sense consciousnesses (which are non-conceptual). The conceptual universe corresponds to the sixth (mental) and seventh (kleshic) consciousness. The seventh (kleshic) consciousness is both conceptual and polluted by incorrect (selfish) understanding, so it is no longer operative in the minds of Arhats. The the eighth consciousness exists as a substrate for mind in general, and acts as everything from the Buddhist perspective.

The relationship between the perceptual and the conceptual universes also corresponds closely with Tibetan Buddhist (and particularly Gelukba) theories of direct and subsequent cognizers. Similarly, the differentiation of concepts into first-order and higher-order corresponds to the distinction between meaning generalities and term generalities (*don spyi* and *gras spyi*).

Cognitive Set Theory is a view of human *cognition*: it does not discuss the role of affect, emotions, and love. Thus, in comparison to Buddhism, it does not attempt to explain the role of the heart (which is probably more important, but more difficult to understand). Neither does it discuss the path which leads away incorrect conceptualization (and negative emotions) and toward a more healthy state of being. These are, of course, the rationale for correctly understanding reality in the first place, so hopefully the conceptual framework in this book leads toward a soteriogical transformation. May it benefit all beings.

This two-fold division between matter and mind is called Cartesian dualism in western philosophy, although it exists in some of the oldest Indian texts as the division between name (nama) and form (rupa). In distinction to classical dualism, Cognitive Set Theory introduces a third universe, by dividing the mental universe into two further universes: the perceptual and the conceptual (which makes obvious sense, since the perception of a thing is not identical to the concept of a thing). This distinction also exists at least as far back as the Buddhist Sautrantika school, and possibly much earlier.

Each of these universes contains or mirrors the others, depending on one’s perspective. As an example of containment, the perceptual world of an individual exists within the physical world: this is how we (in the west) most often understand things. However, the physical world is perceived by an individual, and in that way it exists within the perceptual world. Each of these perspectives is correct from its own perspective.

An example of the mirroring that occurs between these universes is how we typically assume that our perception mirrors physical reality. The division of reality into multiple layers which mirror one was also established in early Indian philosophy if not before: Samkya philosophy uses a double-sided mirror as an analogy for consciousness, which reflects both external (form) and psychological (name) worlds. In contrast to Samkya philosophy, Cognitive Set Theory is explained primarily from the view of scientific realism. This does not exclude other perspectives: hopefully, it can serve as an initial view from which to understand more spiritual dimensions of experience.

The picture below depicts a referential relationship between these universes: the objective (U), perceptual (O), and conceptual (V):

A referential (mirroring) relationship is depicted between these three universes. In accordance with the common view of things, the physical world is primary, and perceptual references to that world are secondary. Things in the physical world are called “objects”, and the things that reference them are called “percepts”. Naturally enough, conceptual references to those percepts are called “concepts”.

This analysis of reality is a fairly simple decomposition of something which is not ultimately divided. This decomposition does not entail a permanent fissure in reality, but rather points to different referential layers, which ultimately exist in accord with their referents to the degree that they are accurate reflections.

As mentioned, these three ontological layers are found in several Buddhist theories (ontological because Buddhists most often explain things using non-conceptual mind (perception) as the basis of experience). The Dalai Lama characterizes them as follows (The Universe in a Single Atom, p.125):

- Matter – physical objects
- Mind – subjective experiences
- Abstract composites – mental formations

More extensive elaborations of the connections between CST and Buddhism can be found in subsequent posts.

This three-fold division of reality into physical, perceptual, and conceptual spheres occurs in numerous non-Buddhist contexts. In psychology, this mental division exists in numerous forms, such as the division between episodic and semantic memory. Karl Popper’s three-fold division of reality characterizes these three divisions as *worlds* (although his world three corresponded to abstract thoughts made incarnate in the world as opposed to abstract thoughts with a mental basis).

In any case, we do not regard dualism to be a very refined point of view. It would be much more descriptive to talk about reality in terms of multiple layers, and it would be closer to the truth to talk in non-dual terms (which turns out to be both an amusing and elusive project). Hopefully, the three universes in Cognitive Set Theory offer a step in this more refined direction.

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