Sameness of qualia

Distinctions among the different levels of sameness (identity > isomorphic > equivalent > adjunction > functor) introduced in category theory is so useful. The more I read philosophical discussion on qualia, I appreciate these distinctions…

The above is Fig 1B of our paper : Tsuchiya, Taguchi, Saigo 2016 Neurosci Research. I’m curious how much people care about the “strength” of the concept of “sameness”. To be honest, I probably didn’t pay much attention to the issue of “sameness” until I met with Hayato Saigo and Shigeru Taguchi in 2015.

Goodman 1977 says “we need only recognize that two qualia are identical if and only if they match all the same qualia.”

What does this sentence mean?

Imagine we are talking about a psychophysical experiment, where qualia of two patches are compared. Then, necessarily, the two patches are already different in space or time.

This means then if we consider “spacetime” relation to consider a category of qualia, these two qualia can’t be “identical” in any way.

But, if “theorists” decide to ignore these relations, then, two qualia can be claimed to be “identical” (but as one is already distinguishing two in some way, it doesn’t make sense to talk about something like this….)

In most cases, what is meant by “identical” is probably at most “isomorphic”. It’s probably more likely to be “equivalent”. Unknowingly, we tend to actively discard what are different when we compare two things other than space and time. For example, unless we ignore textures, shapes, shapes, tastes, etc no two apples can be the same.

Maybe this state of affair continues unless math education entirely rests on a set theoretic notion. In category of Sets, we don’t care about internal relations among the members (=discrete). So as long as the members are the same, two sets are isomorphic.

Let’s say 2 persons have qualia structures with 3 qualia. If their internal relational structures are different, their structures can’t be the same.

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8月の池上高志さん・石黒浩さんとの意識対談と自然変換について考えた

８月に突然、２回Youtubeライブで東大の池上高志さん https://youtu.be/KkQqwiFW8p4 と、阪大・ATRの石黒浩 さんhttps://youtu.be/Y6Pyns3O1vg と対談をさせてもらう機会に恵まれた。あれから２ヶ月ちょっとで両方とも５００回以上再生されている。

お二人とは前にも話したことあったが、今回は、私が圏論の米田の補題の世界観をもって話したから話が深まった気がする。（米田ペーパー日本語版・英語版）

対談の前に、池上さんと石黒さんの共著「人間と機械のあいだ」を読んで準備したのも良かったのだろう。本を読んでいたときにとっていたノートを今見返してみると、池上さんの人工生命とか、石黒さんのロボットとか、普段相手にしている「意識と脳の関係性」というテーマから離れているからこそ見えてくる「共通の構造」的なものに反応していることに気付いた。

この違うものの間の共通性、というのは圏論でいう「自然変換 / natural transformation」に通じるものがある。よく、圏論を振りかざしている人はカッコつけてるだけで中身がない、と言うことをいう人がいる。だが、この自然変換・natural transformation という概念こそは、圏論によってはじめてちゃんと定式化されたものだ。圏論を学び始めた当初、西郷さんに繰り返し、「自然変換が大事」と言われ続けたが、腑に落ちるまでは時間がかかった。それほど、私にとって新しい概念だったのだと思う。

この自然変換的な視点を持ったおかげで、分野横断型・異分野協働・異分野融合の研究の必要性・利点を、おぼろげながら形式化できそうな気がしている。（この辺は１１月１８日にArayaのHiroHamadaさんとやる意識ラジオで話すことになると思う。）

自然変換を理解することで、「共通構造」を見つけるセンスが磨かれるのでは無いか？ ただし、後で説明するように、アナロジー・メタファーを感じるセンスは、ほとんどすべての人がもとから持っている。

「人工生命A とロボットR と人間H が X という共通構造を持つ」とはどういうことか？ たとえば、Xを視覚、としてみる。Aのなかでの視覚の決まり、プロパティ、機能などなどをまとめ上げて作り上げる Aの視覚の圏を想定する。「圏（けん）」についてのイントロは色んな本か、私が西郷さんと書いたものを参照してください（後者は意識研究者向けに書いたのでやさしいはず）。今はグラフみたいなものと考えてもらっていいです。そしてそれがRの視覚カテゴリやHの視覚カテゴリに構造を壊すことなく写せるとしよう。

これのお互いの構造を壊すことなく写せる、というのが、A,R,Hの視覚の間になんらかの「関手（かんしゅ）, functor」がある、という状況だ。（関手についても西郷さんと書いたものを参照）。以下はFig3 from Tsuchiha & Saigo 2021

そしてそういう異なった関手の間に共通する法則性というか、翻訳の決まり、みたいなメタなレベルでの共通構造がある時、それが 「視覚」というレベルにおける「自然変換」がある、という状況だ。以下はFig5 from Tsuchiha & Saigo 2021

ただし、ここまでだけだと、なんだ、アナロジー・メタファーとどう違うんだ？という話になる。圏論とかいってカッコつけんな、的な。

しかし、ここからが最初、私の腑に落ちなかった話しになってくる。

自然変換は、ただのアナロジー・メタファーではない。「自然」だとみなせるのはどういうとき、というのがきっちり数学的に定義されている。関手が移した先の圏の間における関係性（=arrow, 射）の集合としてCoherentになっていなければならない。tA, tB, …. というのが上のFig 5における射の集合、自然変換になっている。この縛りがキツイ。きついがガチガチというわけではない。というのも、もともとの圏における関係性として本当にA,R,Hが「自然」と似ているんであれば、そこは当然満たされるべき条件だからだ。

そして、圏論のすごいところは、圏、関手、自然変換とメタレベルが上がりつづけるように思えるが、自然変換のレベルで一回、圏における射の集合というレベルに話が落ち着くところだ。これは味わい深い。

我々の論文では、自然変換が成り立っている可能性が高い例として、視野のどこにも写せるような共通の構造を挙げている。右視野だろうと、左視野だろうと、中心窩（Fovea)だろうと、周辺視野だろうと、写せるそういう構造。移した後にローカルな関係性が保たれるような辻褄があうような「自然な関係性」。

で、ここまで考え抜くと、そんなに簡単にA,R,Hの間に自然変換は見つからないだろうということがわかる。

ロボット・人工生命・人間の違いに目を向けるのは簡単だ。何が同じか、同じものを作るにはどうしたらいいか、を考えるのは難しい。

Why do we need to care category theory and quantum cognition?

In my latest Consciousness Dialogue with Alex Maier [Link], @alexvmaier we chatted a bit about this issue.

Why am I motivated to learn and apply category theory on the data in consciousness research? Why am I interested in quantum cognition by Busemeyer, Bruza, Trueblood, @EmmanuelPothos , ….?

In our discussion, I pointed out an empirical need for a theory that can explain enigmatic findings in similarity experiments. In most similarity experiments, we (implicitly) assume that inverse of similarity (=dissimilarity) is like a distance. A metric (or distance) needs to satisfy the following three axioms. https://en.wikipedia.org/wiki/Metric_(mathematics)

1. Minimality: d(a,b) = 0 <=> a = b
2. Symmetry: d(a,b) = d(b,a)
3. Triangle inequality: d(a,b) <= d(a,c) + d(c,b)

In the video, we chatted a bit on the violation of “2. symmetry”. We are preparing several drafts on this. If you are interested, please see Tversky 1977, Pothos et al 2013 Psych Review.

There is one thing that I couldn’t mention in the video. And in fact, this may be more important. We need a “theory” that is “independent” of the data. This is most eloquently described Steve Phillips’s paper in 2010 (PLoS Comp)

“The Ptolemean (geocentric) theory’s additional assumption (called “epicycles”) is ad hoc. It is unconnected with the rest of the theory and motivated only by the need to fit the data—the assumption could not be confirmed independently of confirming the theory.”

Category theory is a mathematical theory constructed to study “structures”.

Quantum cognition is a mathematical theory that is based on noncommutative probability theory.

These theories are built independent of the data that we want to explain (e.g., the properties of consciousness / qualia). We don’t know if it’s applicable to the data or not.

This is different from testing “theories” that were constructed from the data to be explained. Such “theories” is unfortunately immune to “ad hoc” explanation. Even the theorists themselves do not notice how the theory has changed over the time “to fit” it with the data….

On the process theory and qualia

On Picturing Quantum Processes by Coecke & Kissinger

David Bohm and David Peat wrote in 1987 “We haven’t actually paid much attention to thought as a process. We have engaged in thoughts, but we have only paid attention to the content, not to the process”

I’m guessing that the same may be true for qualia / consciousness.

Here “process” means what is discussed in “process theory”. Process is anything that has zero or more inputs and zero or more outputs. Process theories seem quite useful in the context of quantum theory (and it has a link to category theory).

A function takes one or more inputs and outputs one. Functions are just one example of processes.

A more general notion is a relation. A relation links many inputs to many outputs. It can be considered as a stochastic output. Relations are much closer to what neurons are and more likely to be useful to understand how the brain works. (But most neuroscientists including myself probably don’t know much about relations…)

Ch3 of the Picturing Quantum Processes explains it very well “3.3 Functions and Relations as Processes”.

And processes are even more general. It allows even NO inputs or NO outputs. And this is the reason I think processes are much better concepts than functions or relations when we think about qualia and consciousness.

Thinking about consciousness as something that is defined / induced / evoked by stimulus is misleading. Something that causes motor output is also misleading. Consciousness exists in dreaming and hallucination. Consciousness exists in locked-in or even more severe brain damaged patients. Phenomenal consciousness may or may not be reportable. But if phenomenal consciousness is considered as a process, it doesn’t matter.

Our latest paper finally came out! The Yoneda lemma paper!

Naotsugu Tsuchiya, Hayato Saigo, A relational approach to consciousness: categories of level and contents of consciousness, Neuroscience of Consciousness, Volume 2021, Issue 2, 2021, niab034, https://doi.org/10.1093/nc/niab034

Finally, our paper is out! In this paper, we introduce one of the most important consequence in category theory, the Yoneda Lemma, to consciousness research! @FQXi

The Yoneda perspective, applied to consciousness research, implies that we can characterize any quale through characterizing a massive relationships between a quale and other qualia! In math terms, hA~=hB <-> A~=B.

Along the way to get to the Yoneda lemma, we gently introduce various useful concepts in category theory (e.g., category, functor, natural transformations, hom functors, equivalence). This paper can serve as an entry point to category theory for consciousness researchers!

This paper will be the first part of our trilogy. The second one (under revision) will extend this approach to “enriched category theory”. With that, we can deal with the graded levels of relations between qualia (e.g., similarity ratings).

This will allow many important theoretical tools in category theory to connect with empirical psychophysics, neuroscience (e.g., representational similarity) and theoretical research (e.g., maximally irreducible conceptual structure in the integrated information theory).

(The last one (which we haven’t started writing…) is supposed to close any loose end through sheaf theory.) Anyways, I didn’t know that this paper was already published already last week…