It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.3K
- Chat 500
- Study Groups 19
- Petri Nets 9
- Epidemiology 4
- Leaf Modeling 1
- Review Sections 9
- MIT 2020: Programming with Categories 51
- MIT 2020: Lectures 20
- MIT 2020: Exercises 25
- MIT 2019: Applied Category Theory 339
- MIT 2019: Lectures 79
- MIT 2019: Exercises 149
- MIT 2019: Chat 50
- UCR ACT Seminar 4
- General 68
- Azimuth Code Project 110
- Statistical methods 4
- Drafts 2
- Math Syntax Demos 15
- Wiki - Latest Changes 3
- Strategy 113
- Azimuth Project 1.1K
- - Spam 1
- News and Information 147
- Azimuth Blog 149
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 713

Options

Chris Goddard, I saw your comment and your link to your recent paper, A map of a research programme for subtlety theory. At Research Gate, I also found your book, A treatise on information geometry. and your papers on game theory.

Would you like to exchange ideas related to subtlety theory and information theory? I am curious how you look at these subjects. Perhaps we have overlapping interests. Could would explore them here?

## Comments

I noticed that in your paper on subtlety theory you mention Roy Frieden's book, Physics from Fisher Information.

I read a blogger's skeptical review of Frieden's book. Perhaps you might address those criticisms from your own perspective. Or, more importantly, to explain what you find compelling, or simply to write more about your personal goals, in general.

`I noticed that in your paper on subtlety theory you mention Roy Frieden's book, <a href="http://catdir.loc.gov/catdir/samples/cam032/98020461.pdf">Physics from Fisher Information.</a> I read a blogger's <a href="http://bactra.org/reviews/physics-from-fisher-info/">skeptical review</a> of Frieden's book. Perhaps you might address those criticisms from your own perspective. Or, more importantly, to explain what you find compelling, or simply to write more about your personal goals, in general.`

I would be curious to learn more about Fisher information for the following reason.

Here in Lithuania, Šarūnas Raudys won a national science award for his work on classifiers in statistics and neural networks. He wrote a book, Statistical and Neural Classifiers: An Integrated Approach to Design He describes the following hierarchy of classifiers:

His main conclusion is that it is very important, in training neural networks, that they not overlearn. Which is to say, there always needs to be some noise in the training so that they stay open to new types of learning, which can be more sophisticated, as per this hierarchy. If they overlearn, then they will be stuck in a rut, and won't be able to rise to a higher level of sophistication.

I was intrigued that his hierarchy might relate to the building blocks of my philosophy, which are cognitive frameworks that I call "divisions of everything". I write about them in my presentation, Time and Space as Representations of Decision-Making. My thought was that these classifiers might be distinguished by an increasing number of perspectives, from one to seven. I visited him a few times but I didn't manage to interest him in this and so I left it at that.

In recent years, I have been looking for similar frameworks in mathematics, and thus have been studying Bott periodicity and Clifford algebras, because they manifest an eight-cycle, as do the divisions of everything. It seems plausible that these structures express what happens as we add perspective after perspective. In particular, n-spheres may be related to assemblies of perspectives.

I have a B.S. in Physics and a Ph.D. in Math. So I know some basics of statistics from my Physics courses. But I would have to study the classifiers to understand what they are and what they do. But they generally same to be quadratic functions. So I thought they might be related to Clifford algebras. I am curious if you have thoughts about Clifford algebras and/or geometric algebra.

Would any of this be of interest to you? And why? I am curious.

`I would be curious to learn more about Fisher information for the following reason. Here in Lithuania, Šarūnas Raudys won a national science award for his work on classifiers in statistics and neural networks. He wrote a book, <a href="https://books.google.lt/books?id=W94LBwAAQBAJ&printsec=frontcover&hl=lt#v=onepage&q&f=false">Statistical and Neural Classifiers: An Integrated Approach to Design</a> He describes the following hierarchy of classifiers: <ol><li>the Euclidean distance classifier; <li>the standard Fisher linear discriminant function (DF); <li>the Fisher linear DF with pseudo-inversion of the covariance matrix; <li>regularized linear discriminant analysis; <li>the generalized Fisher DF; <li>the minimum empirical error classifier; <li>the maximum margin classifier. </ol> His main conclusion is that it is very important, in training neural networks, that they not overlearn. Which is to say, there always needs to be some noise in the training so that they stay open to new types of learning, which can be more sophisticated, as per this hierarchy. If they overlearn, then they will be stuck in a rut, and won't be able to rise to a higher level of sophistication. I was intrigued that his hierarchy might relate to the building blocks of my philosophy, which are cognitive frameworks that I call "divisions of everything". I write about them in my presentation, <a href="http://www.ms.lt/sodas/Book/20170929TimeSpaceDecisionMaking">Time and Space as Representations of Decision-Making</a>. My thought was that these classifiers might be distinguished by an increasing number of perspectives, from one to seven. I visited him a few times but I didn't manage to interest him in this and so I left it at that. In recent years, I have been looking for similar frameworks in mathematics, and thus have been studying Bott periodicity and Clifford algebras, because they manifest an eight-cycle, as do the divisions of everything. It seems plausible that these structures express what happens as we add perspective after perspective. In particular, n-spheres may be related to assemblies of perspectives. I have a B.S. in Physics and a Ph.D. in Math. So I know some basics of statistics from my Physics courses. But I would have to study the classifiers to understand what they are and what they do. But they generally same to be quadratic functions. So I thought they might be related to Clifford algebras. I am curious if you have thoughts about Clifford algebras and/or geometric algebra. Would any of this be of interest to you? And why? I am curious.`

Hi @AndriusKulikauskas,

Apologies for the delay in my reply. I have not been frequenting this forum of late. If you tag me e.g. @ChrisGoddard, I should get an email notification though.

Thank you for your interest in my work.

Certainly, I would certainly be interested in exploring the possibility of dialogue.

There have been a number of parties who have written critically about Roy Frieden's work. One notable example of this is a more detailed critique of Frieden's book here which is probably a bit more detailed in terms of its analysis of Frieden's ideas than the blog post you linked above. At the end of this article, D.A. Lavis and R.F. Streater end with the remark (emphasis mine):

I would consider my research programme that builds on top of Roy Frieden's research programme as such a potentially "new and interesting work" which has been stimulated by Roy Frieden's work on the matter. In such a way my work is neither in conflict with the view of Lavis / Streater, nor of Frieden, but rather in a sense orthogonal to these other two parties.

In terms of the manner in which the polemic was written, I have little to comment, other than the fact that I think a little bit of humility and consideration can go a long way in general when commenting on the work of others. People should really try to build each other up in the sciences, rather than tear one another down.

Returning to the blog post you mentioned above - my key difficulty with Cosma's blog post is that it is written more from the perspective of the philosophy of physics, and therefore by its very nature is inherently woollier than simply analysing the actual mathematics of what Roy Frieden was doing. It is also not a paper, but a blog post, and therefore should not be read as a serious critique, in the same way that, say, a paper would be. Finally, I am not overly interested in philosophy of physics (although I can understand its value); rather I tend to be more interested in the "shut up and compute" philosophy as extolled I think by at least one prominent academic in recent memory.

What I found compelling about Roy Frieden's work was that it provided a potential avenue to understanding how to derive hamiltonians for physics from first principles. Even if his approach was not thoroughly rigorous and mathematically watertight, I think from an intuitionistic perspective he was clearly onto something. That is what drew me to his research originally in 1999, and pulled me back after I had learned a bit more physics in 2004 and was about to start my doctoral studies.

Understanding where hamiltonians come from or having some idea as to a principle as to how to do so would be potentially extremely significant and a very powerful insight. Indeed, this was hammered home to me from my undergraduate and honours studies - it seemed to me that on the basis of 19th and 20th century physics that there was really no clear explanation or consensus as to where really action functionals came from, other than through the use of tried and tested heuristics and "physical intuition" based loosely on arguments of "symmetry".

(The latter of these - appeals to notions of symmetry - presumably relates to thinking of things in terms of Lie Groups that act on geometric models in which dynamics occurs, and in a way could potentially be made more rigorous in and of itself, however this was not an avenue of thought towards which my interests took me. Needless to say, however, I have my doubts that the existing machinery presents an acceptable level of generality.)From a mathematician's perspective, however, appealing vaguely to "physical intuition" and "symmetry" (in a primitive fashion) is not good enough - one would really like to be able to "derive" or "prove" that the equations of electromagnetism, general relativity, quantum mechanics etc can be derived from particular natural assumptions about the geometry in which dynamics is to take place. Frieden laid the groundwork there, and I'd like to think that I did a fair bit of work from 2005 to 2010 fleshing it out, and then from 2010 to 2020 in terms of generalising this framework.

My personal goals in terms of my research programme at present are to try to hand over a bit of what I've done. I think that this is reasonable, as I am no longer a young man, and am quite definitely middle-aged now. Also, I need a bit of a break from thinking about some of these things.

There are obviously threads of thought I'd like to potentially pick up again at some point, maybe in another three to five years or so, but I'm in no hurry to continue doing serious research at the moment; particularly since my paid work at present is as an engineer (which I enjoy tremendously, and have no real desire to change) rather than as an academic.

I have been attracted to this forum again really for two reasons:

I have things to do now, but I haven't neglected to notice your last comment about your own research. I will draft and post another comment in the next day or so sharing my thoughts on same. Here though is a teaser - I am reminded from having a quick view of your own work of Dave Snowden's Cynefin framework.

`Hi @AndriusKulikauskas, Apologies for the delay in my reply. I have not been frequenting this forum of late. If you tag me e.g. @ChrisGoddard, I should get an email notification though. >Chris Goddard, I saw your comment and your link to your recent paper, A map of a research programme for subtlety theory. At Research Gate, I also found your book, A treatise on information geometry. and your papers on game theory. Thank you for your interest in my work. >Would you like to exchange ideas related to subtlety theory and information theory? I am curious how you look at these subjects. Perhaps we have overlapping interests. Could would explore them here? Certainly, I would certainly be interested in exploring the possibility of dialogue. >I noticed that in your paper on subtlety theory you mention Roy Frieden's book, [Physics from Fisher Information](catdir.loc.gov/catdir/samples/cam032/98020461.pdf). > >I read [a blogger's skeptical review of Frieden's book](http://bactra.org/reviews/physics-from-fisher-info/). Perhaps you might address those criticisms from your own perspective. [...] There have been a number of parties who have written critically about Roy Frieden's work. One notable example of this is a more detailed critique of Frieden's book [here](http://faculty.poly.edu/~jbain/physinfocomp/Readings/02LavisStreater.pdf) which is probably a bit more detailed in terms of its analysis of Frieden's ideas than the blog post you linked above. At the end of this article, D.A. Lavis and R.F. Streater end with the remark (emphasis mine): >"We regret to say that we find this book to be fundamentally flawed in both its overall concept and mathematical detail. It cannot be read as a textbook providing a valid approach to physics. *However it could, perhaps, be a source of stimulation for some new and interesting work.*" I would consider my research programme that builds on top of Roy Frieden's research programme as such a potentially "new and interesting work" which has been stimulated by Roy Frieden's work on the matter. In such a way my work is neither in conflict with the view of Lavis / Streater, nor of Frieden, but rather in a sense orthogonal to these other two parties. In terms of the manner in which the polemic was written, I have little to comment, other than the fact that I think a little bit of humility and consideration can go a long way in general when commenting on the work of others. People should really try to build each other up in the sciences, rather than tear one another down. Returning to the blog post you mentioned above - my key difficulty with Cosma's blog post is that it is written more from the perspective of the philosophy of physics, and therefore by its very nature is inherently woollier than simply analysing the actual mathematics of what Roy Frieden was doing. It is also not a paper, but a blog post, and therefore should not be read as a serious critique, in the same way that, say, a paper would be. Finally, I am not overly interested in philosophy of physics (although I can understand its value); rather I tend to be more interested in the "shut up and compute" philosophy as extolled I think by at least one prominent academic in recent memory. >[...] Or, more importantly, to explain what you find compelling [...] What I found compelling about Roy Frieden's work was that it provided a potential avenue to understanding how to derive hamiltonians for physics from first principles. Even if his approach was not thoroughly rigorous and mathematically watertight, I think from an intuitionistic perspective he was clearly onto something. That is what drew me to his research originally in 1999, and pulled me back after I had learned a bit more physics in 2004 and was about to start my doctoral studies. Understanding where hamiltonians come from or having some idea as to a principle as to how to do so would be potentially extremely significant and a very powerful insight. Indeed, this was hammered home to me from my undergraduate and honours studies - it seemed to me that on the basis of 19th and 20th century physics that there was really no clear explanation or consensus as to where really action functionals came from, other than through the use of tried and tested heuristics and "physical intuition" based loosely on arguments of "symmetry". *(The latter of these - appeals to notions of symmetry - presumably relates to thinking of things in terms of Lie Groups that act on geometric models in which dynamics occurs, and in a way could potentially be made more rigorous in and of itself, however this was not an avenue of thought towards which my interests took me. Needless to say, however, I have my doubts that the existing machinery presents an acceptable level of generality.)* From a mathematician's perspective, however, appealing vaguely to "physical intuition" and "symmetry" (in a primitive fashion) is not good enough - one would really like to be able to "derive" or "prove" that the equations of electromagnetism, general relativity, quantum mechanics etc can be derived from particular natural assumptions about the geometry in which dynamics is to take place. Frieden laid the groundwork there, and I'd like to think that I did a fair bit of work from 2005 to 2010 fleshing it out, and then from 2010 to 2020 in terms of generalising this framework. >[...] , or simply to write more about your personal goals, in general. My personal goals in terms of my research programme at present are to try to hand over a bit of what I've done. I think that this is reasonable, as I am no longer a young man, and am quite definitely middle-aged now. Also, I need a bit of a break from thinking about some of these things. There are obviously threads of thought I'd like to potentially pick up again at some point, maybe in another three to five years or so, but I'm in no hurry to continue doing serious research at the moment; particularly since my paid work at present is as an engineer (which I enjoy tremendously, and have no real desire to change) rather than as an academic. I have been attracted to this forum again really for two reasons: * one, because there are top mathematicians and scientists who occasionally visit this forum or its loosely affiliated entity, the n-category cafe and the n-lab, who might be able to accept the messy bundle of insights that I've gathered piecemeal over the last 15 years, in a part-time fashion - and which is summarised as you mentioned in [this paper](https://www.researchgate.net/publication/338825550_A_map_of_a_research_programme_for_subtlety_theory), * and two, because the ideas being taught here at present regarding functional programming with categories fit my current agenda of becoming a better programmer / software engineer - which is my current chosen vocation. >[My motivation to learn more about the Fisher Information] I have things to do now, but I haven't neglected to notice your last comment about your own research. I will draft and post another comment in the next day or so sharing my thoughts on same. Here though is a teaser - I am reminded from having a quick view of your own work of [Dave Snowden's Cynefin framework](https://en.wikipedia.org/wiki/Cynefin_framework).`

Thanks for the pointer to the book of Š. Raudys. Machine learning is something that does interest me, but it is not a field that I currently work in. However that work does look like a very good piece of scholarship on pattern recognition - and it does make use of local measures of information, such as the Fisher Information functional. If I manage to find the time, maybe I'll seek to borrow a copy from a university library at some point (after the present chaos is over, of course), and have a bit of a read.

I haven't heard of this sort of idea before but it sounds interesting, and potentially natural from the point of view of machine learning. Certainly when building convolutional neural networks for image recognition, lower networks learn lines and shapes first, before the network abstracts naturally to more complicated artifacts, depending on the training data. Again, I'm not an expert, but I've had the privilege of rubbing shoulders with a number of people at my office trained in this sort of thing, so I've learned a little through osmosis, so to speak.

Thanks for sharing your earlier work. I've had a quick glance through what you've written.

In terms of a seven fold hierarchy of classifiers, I think that there might be potentially a relationship with the idea of Lens Categories. I am reminded of a conversation that I had with someone (a Professor of Biomedical Engineering) in the states in early 2019. His question was really an optics question, but nonetheless there does seem to be a potential link here. After a bit of a chat, I thought that what he mentioned seemed to have some relation to this 1979 paper about the elliptic umbilic diffraction catastrophe. A more modern variant of same establishing an interesting connection with Airy rings can be found here in this paper. In particular there seems to be some tie in between categories that have a lens space topology, between optics, and between catastrophe theory. I have not explored these connections very thoroughly.

Nonetheless I think this is a bit tangential to your interests and maybe totally off-track.

Yes, so as mentioned, I had a look at your work. A number of questions and observations come to mind.

One set of questions is: What is significant about a hierarchy of classifiers? How would this generalise to other problems apart from pattern recognition?

Another would be: Why seven? Where does this number come from? Can we construct this from a more primitive framework in terms of which a hierarchy of classifiers and their multiplicity are emergent properties?

One observation is that your work might benefit a little from asking yourself questions like: can this be made simpler? How can I express this more mathematically? Can I pare away unnecessary words and concepts here? Which ideas are essential and which are just labels?

Also,

Can I rigorously prove any of the statements in my programme? If I wanted to prove any of these statements, how would I go about that?

I hope you don't mind all of the questions I have posed above. Hopefully they will help to sharpen your ideas.

Nonetheless, I have to admit I am drawn to this emergent property of the work of Raudys, and am intrigued by this particular set of questions that I asked above:

There is certainly an intriguing mystery here. However a blunt attack on understanding this sort of thing might not succeed; one would need to find an oblique way to try to dig a bit deeper into some of the structure here.

`>Here in Lithuania, Šarūnas Raudys won a national science award for his work on classifiers in statistics and neural networks. He wrote a book, [Statistical and Neural Classifiers: An Integrated Approach to Design](https://www.amazon.com.au/Statistical-Neural-Classifiers-Integrated-Approach/dp/1852332972). Thanks for the pointer to the book of Š. Raudys. Machine learning is something that does interest me, but it is not a field that I currently work in. However that work does look like a very good piece of scholarship on pattern recognition - and it does make use of local measures of information, such as the Fisher Information functional. If I manage to find the time, maybe I'll seek to borrow a copy from a university library at some point (after the present chaos is over, of course), and have a bit of a read. >He describes the following hierarchy of classifiers ... His main conclusion is that it is very important, in training neural networks, that they not overlearn. Which is to say, there always needs to be some noise in the training so that they stay open to new types of learning, which can be more sophisticated, as per this hierarchy. I haven't heard of this sort of idea before but it sounds interesting, and potentially natural from the point of view of machine learning. Certainly when building convolutional neural networks for image recognition, lower networks learn lines and shapes first, before the network abstracts naturally to more complicated artifacts, depending on the training data. Again, I'm not an expert, but I've had the privilege of rubbing shoulders with a number of people at my office trained in this sort of thing, so I've learned a little through osmosis, so to speak. >I was intrigued that his hierarchy might relate to the building blocks of my philosophy, which are cognitive frameworks that I call "divisions of everything". I write about them in my presentation, [Time and Space as Representations of Decision-Making](http://www.ms.lt/sodas/Book/20170929TimeSpaceDecisionMaking). My thought was that these classifiers might be distinguished by an increasing number of perspectives, from one to seven. Thanks for sharing your earlier work. I've had a quick glance through what you've written. In terms of a seven fold hierarchy of classifiers, I think that there might be potentially a relationship with the idea of Lens Categories. I am reminded of a conversation that I had with someone (a Professor of Biomedical Engineering) in the states in early 2019. His question was really an optics question, but nonetheless there does seem to be a potential link here. After a bit of a chat, I thought that what he mentioned seemed to have some relation to this [1979 paper about the elliptic umbilic diffraction catastrophe](https://www.researchgate.net/publication/252379656_The_Elliptic_Umbilic_Diffraction_Catastrophe). A more modern variant of same establishing an interesting connection with Airy rings can be found [here in this paper](https://www.researchgate.net/publication/231110105_From_Airy_rings_to_the_elliptic_umbilic_diffraction_catastrophe). In particular there seems to be some tie in between categories that have a lens space topology, between optics, and between catastrophe theory. I have not explored these connections very thoroughly. Nonetheless I think this is a bit tangential to your interests and maybe totally off-track. >In recent years, I have been looking for similar frameworks in mathematics, and thus have been studying Bott periodicity and Clifford algebras [...] Yes, so as mentioned, I had a look at your work. A number of questions and observations come to mind. One set of questions is: What is significant about a hierarchy of classifiers? How would this generalise to other problems apart from pattern recognition? Another would be: Why seven? Where does this number come from? Can we construct this from a more primitive framework in terms of which a hierarchy of classifiers and their multiplicity are emergent properties? One observation is that your work might benefit a little from asking yourself questions like: can this be made simpler? How can I express this more mathematically? Can I pare away unnecessary words and concepts here? Which ideas are essential and which are just labels? Also, Can I rigorously prove any of the statements in my programme? If I wanted to prove any of these statements, how would I go about that? >[Hierarchy of classifiers] I hope you don't mind all of the questions I have posed above. Hopefully they will help to sharpen your ideas. Nonetheless, I have to admit I am drawn to this emergent property of the work of Raudys, and am intrigued by this particular set of questions that I asked above: * Why seven? Where does this number come from? * Can we construct this from a more primitive framework in terms of which a hierarchy of classifiers and their multiplicity are emergent properties? There is certainly an intriguing mystery here. However a blunt attack on understanding this sort of thing might not succeed; one would need to find an oblique way to try to dig a bit deeper into some of the structure here.`

Chris, thank you for responding. It's a privilege to get your letters.

As regards my own work, I'm not trying to prove anything. Rather, I'm trying to document and understand the conceptual frameworks that limit my thinking, that my thinking leverages and that express everything I could possibly know.

I visited Dr.Raudys several times and more than ten years ago I asked him to explain to me, intuitively, each of the classifiers, and I recall that, as I hypothesized, they did seem to break down the space with the help of an increasing number of conceptual parameters. But I think we need to add a zeroth possibility.

In my study of the divisions of everything, what happens that keeps them from growing infinitely is that we get the logical square with four corners and three sides that defines a self-standing logical system. But if we add the eighth side, say "All are good AND All are bad", then the system must be empty, in which case it all collapses into the state of contradiction. Thus the division of everything into eight perspectives is the same as the division of everything into no perspectives.

I have worked on such questions since 1982 and in 2014-2019 gave about 40 talks in English and Lithuanian on various aspects. Now I have savings that allow me to work on this for at least two years. So this year I am finishing my research and next year I plan to write a series of academic papers.

Currently, in my research, I am modeling what it means to go through a life experience. I am also spending a lot of time to investigate and understand the whole of math. My frameworks are helping me to zero in on what I think are the key structures in mathematics, the belly button, so to speak. Here is my mathematical notebook. I think that I will be able to get lots of papers published in my philosophy, and that will help validate my thinking, but nobody will care. But if I can express my ideas mathematically, and also explain how all of math unfolds conceptually, with its ideas, branches, questions and results, and perhaps get some new math results, too, then I think that people will try to reverse engineer what I have done, and try to apply these ideas.

One nice example is my talk A Geometry of Moods: Evoked by Wujue Poems of the Tang Dynasty. I document six transformations and just last year I realized that four of them (reflection, rotation, dilation, translation) are Mobius transformations, which is SU(2), and the other two (shear, squeeze) appear inside the 2x2 matrices used to represent one-parameter subgroups of SU(2) with 2x2 matrices. So this is an example where I am uncovering a new outlook on a mathematical structure (both outside and inside, as it were) which a normal mathematician would never see (because of the mental boundaries they place).

`Chris, thank you for responding. It's a privilege to get your letters. As regards my own work, I'm not trying to prove anything. Rather, I'm trying to document and understand the conceptual frameworks that limit my thinking, that my thinking leverages and that express everything I could possibly know. I visited Dr.Raudys several times and more than ten years ago I asked him to explain to me, intuitively, each of the classifiers, and I recall that, as I hypothesized, they did seem to break down the space with the help of an increasing number of conceptual parameters. But I think we need to add a zeroth possibility. In my study of the divisions of everything, what happens that keeps them from growing infinitely is that we get the logical square with four corners and three sides that defines a self-standing logical system. But if we add the eighth side, say "All are good AND All are bad", then the system must be empty, in which case it all collapses into the state of contradiction. Thus the division of everything into eight perspectives is the same as the division of everything into no perspectives. I have worked on such questions since 1982 and in 2014-2019 gave about 40 talks in <a href="http://www.ms.lt/sodas/Book/Book">English</a> and <a href="http://www.ms.lt/sodas/Mintys/%c4%aevadas">Lithuanian</a> on various aspects. Now I have savings that allow me to work on this for at least two years. So this year I am finishing my research and next year I plan to write a series of academic papers. Currently, in my research, I am modeling what it means to go through a life experience. I am also spending a lot of time to investigate and understand the whole of math. My frameworks are helping me to zero in on what I think are the key structures in mathematics, the belly button, so to speak. Here is my <a href="http://www.ms.lt/sodas/Book/MathNotebook">mathematical notebook</a>. I think that I will be able to get lots of papers published in my philosophy, and that will help validate my thinking, but nobody will care. But if I can express my ideas mathematically, and also explain how all of math unfolds conceptually, with its ideas, branches, questions and results, and perhaps get some new math results, too, then I think that people will try to reverse engineer what I have done, and try to apply these ideas. One nice example is my talk <a href="http://www.ms.lt/sodas/Book/20180815AGeometryOfMoods">A Geometry of Moods: Evoked by Wujue Poems of the Tang Dynasty</a>. I document six transformations and just last year I realized that four of them (reflection, rotation, dilation, translation) are Mobius transformations, which is SU(2), and the other two (shear, squeeze) appear inside the 2x2 matrices used to represent one-parameter subgroups of SU(2) with 2x2 matrices. So this is an example where I am uncovering a new outlook on a mathematical structure (both outside and inside, as it were) which a normal mathematician would never see (because of the mental boundaries they place).`

I am quite the absolute idealist studying the limits of my own mind. But in giving my talks, I realized that I could ground my thinking in a materialist point of view, as I wrote in my talk on Time and Space as Representations of Decision Making. The idea being that resources get divided for dealing with what you know and what you don't know. And more and more, the latter takes over. Let's say there is a zeroth material state, and then a mind that unconsciously knows the answer (Kahneman and Tversky's S1), and a mind that consciously asks questions (S2), and then a consciousness that sets the parameter between the two, whether to step-in or step-out. In this way, the ideal fully emerges from the material. Then this allows for the same extension to be grounded in the opposite direction, from the ideal to the material. This yields another four divisions, bringing us back to where we started. In broad terms, I think that is what should be happening with the hierarchy of classifiers or with the Bott periodicity. But I need to learn the details of such various phenomenon and see if my thinking here is on track or not.

`I am quite the absolute idealist studying the limits of my own mind. But in giving my talks, I realized that I could ground my thinking in a materialist point of view, as I wrote in my talk on <a href="http://www.ms.lt/sodas/Book/20170929TimeSpaceDecisionMaking">Time and Space as Representations of Decision Making</a>. The idea being that resources get divided for dealing with what you know and what you don't know. And more and more, the latter takes over. Let's say there is a zeroth material state, and then a mind that unconsciously knows the answer (Kahneman and Tversky's S1), and a mind that consciously asks questions (S2), and then a consciousness that sets the parameter between the two, whether to step-in or step-out. In this way, the ideal fully emerges from the material. Then this allows for the same extension to be grounded in the opposite direction, from the ideal to the material. This yields another four divisions, bringing us back to where we started. In broad terms, I think that is what should be happening with the hierarchy of classifiers or with the Bott periodicity. But I need to learn the details of such various phenomenon and see if my thinking here is on track or not.`

Chris, I have set up two pages at the wiki for Deriving Hamiltonians and for Hierarchy of Classifiers. I suggest we start new threads on those subjects and whatever is left over could be continued here. I am currently reading the Lavis and Streater review.

Thank you for suggesting the Lens category and for alerting me to the Cynefin framework!

`Chris, I have set up two pages at the wiki for <a href="https://www.azimuthproject.org/azimuth/show/Deriving+Hamiltonians">Deriving Hamiltonians</a> and for <a href="https://www.azimuthproject.org/azimuth/show/Hierarchy+of+Classifiers">Hierarchy of Classifiers</a>. I suggest we start new threads on those subjects and whatever is left over could be continued here. I am currently reading the Lavis and Streater review. Thank you for suggesting the Lens category and for alerting me to the Cynefin framework!`

@ChrisGoddard at this page on Hierarchy of Classifiers I uploaded three papers by Šarūnas Raudys. His book expands on these initial papers. Happy Easter!

`@ChrisGoddard at this page on <a href="https://www.azimuthproject.org/azimuth/show/Hierarchy+of+Classifiers">Hierarchy of Classifiers</a> I uploaded three papers by Šarūnas Raudys. His book expands on these initial papers. Happy Easter!`

Thanks, Happy Easter to you as well!

I don't have a tremendously large amount of bandwidth at present for these matters, but if at some point that changes I will spend a bit of time carefully looking into these matters and attempting to understand how to extract some useful and terse mathematics out of it. Parsimony seems like a good policy, but as a well known writer adroitly pointed out a bit more than a century ago, it is often easier to write a long commentary than a short one. I'd rather aim for quality and brevity here, if possible; and that sort of thing requires effort and time.

`Thanks, Happy Easter to you as well! I don't have a tremendously large amount of bandwidth at present for these matters, but if at some point that changes I will spend a bit of time carefully looking into these matters and attempting to understand how to extract some useful and terse mathematics out of it. Parsimony seems like a good policy, but [as a well known writer adroitly pointed out a bit more than a century ago](https://www.goodreads.com/quotes/21422-i-didn-t-have-time-to-write-a-short-letter-so), it is often easier to write a long commentary than a short one. I'd rather aim for quality and brevity here, if possible; and that sort of thing requires effort and time.`