Not the formal description but the physical brain model is the problem.

If you can do this you will find the corresponding tools in mathematics.

It's hard to tell what pure mathematicians have been doing in the centuries since they began to explore mathematics, but conventional math as used today will probably not do the trick. I am not sure what you call Denotional mathematics though.

Certain mechanisms and/or their aspects can be quite adequately modeled by existing math formalisms. Other mechanisms/aspects might require new, not existing yet formalisms. No doubts that they are to come when the need raises (and is realized), as it has been proven by multiple historical examples. Thus, I support Joseph's opinion that not the formal description is the problem. In the same time, it is naive, IMO, to believe that the physical brain model would be very helpful: would even the full knowledge of a microprocessor physical principles and electronic scheme be anyhow helpful to understand function of an application which runs on it ?..

Nor would the logical description of the core help with software that runs on a network of computers.

While the brain is not as easily understood as hardware and software suggest as a model, there is reason to believe that even network analysis might not be all that helpful in understanding how it works.

However new techniques will certainly not hurt.

There are mechanisms in mathematics that can adequately be employed to model brain mechanism and mental phenomena in cognitive computing and computational intelligence

Dear Henk Smid ! Can the mathematical algorithm to solve every unstructured decision-making problem that is easy for our brain ???

Here are a few "classic" arguments in the Western canon that I was able to gather on the subject over the past years:

1. The cybernetics school (which was supported by A. Turing, W.S. McCulloch, N. Wiener, etc.) in many papers attempt to define human intelligence and decision making through the "neural network" abstraction.

That is a network of nodes that compute on real or rational numbering systems with a weighting and summing function. Although some obstacles came from M. Minsky's remark that XOR could not be learned by the perceptron, the idea is that human decision is perceived as a complex boolean algebra which has an expression in conventional arithmetics.

2. The theoretical computer science approach takes another perspective to the problem by discussing expressive powers of formal systems. This began with B. Russel's Principia Mathematica, K. Gödel's incompleteness theorem and G. Gentzen's inferential logic and followed through Noam Chomsky's discussion on the expressive power of formal languages.

This more or less means that there is a hierarchy of what a given formal system (a brain's decisional system) can do.

Both approaches discussed in 1 and 2, which are closely related to Ancient Oriental philosophy such as the I'Ching or the Vedas, define intelligence "mechanically" or "relatively".

That is, the assumption is made that any physical process that can occur in nature is equivalent to a Turing machine (the extended Church-Turing thesis, Deutsch's thesis). And in that sense, our mind decides the Halting problem when it gets "bored" of solving it perhaps.

I skimmed through your paper and I'll definitely take the time to read into it more in depth, but my first opinion stems from my career as a young computer scientist.

Processes (process algebras, semantics, inference systems) such as those we deal with in business processes have their limits. What I have noticed is that some rules in the business process world, at least here in North America, cannot be denotated or annotated due to constitutional and legal reasons (Reagan's political correctness) and become tacit.

There is then a clear dichotomy between the description language and the procedural reality that arises...

And we are not discussing the complexity issues at hand in a polysemic system as well, which would still be accounted for the denotational mathematics framework you provided in Figure 1.

Mathematics is the abstract science of numbers, quantity, and space as well as their applications in other disciplines of sciences, engineering, societies, and humanities. The basic methodology of mathematics is abstraction and symbolic inferences. The history of sciences and engineering shows that many branches of mathematics have been created in order to meet the abstract, rigorous, expressive, and efficient needs for dealing with the ever increasing complexity in reasoning and problem solving. The phenomenon may be conceived as that new types of problems require new forms of mathematics. It also indicates that the maturity of a new discipline is characterized by the maturity of its mathematical means.

A key practical need in designing and implementing a computational intelligent system or a cognitive computer is how the natural intelligence may be rigorously conveyed to machines. It is obvious that we cannot use pure numbers or binary representations, because it is too insufficient and inefficient. We need suitable mathematical means that would closer to the level of human thinking and inference collectively known as denotational mathematics (DM). DM is a counterpart of conventional analytic mathematics, which were not handy as a means for modeling and generating machine intelligence.

As an example, let’s look at the need for rigorous semantic analysis in natural language processing. There are two categories of semantics in our every day expressions known as the semantics of “to-be” and “to-do” sentences. The former are formally an equivalency between two concepts, while the latter are formally a process. How does existing mathematical means be used to model and analyze these kind of problems?

I have found that the main problem in NLP when comparing it to human intelligence is to be able to weight "What is said" vs "What is not said".

A system such as Google can easily search for documents with good precision over keyword search.

Now, if we take the English nursery rhyme:

Humpty Dumpty sat on a wall,

Humpty Dumpty had a great fall.

All the king's horses and all the king's men

Couldn't put Humpty together again.

Lewis Carroll's book portrays the character Humpty Dumpty as a anthropomorphic egg... (https://www.youtube.com/watch?v=UUrQmfyGgbA)

However, if you take the context in which the rhyme first appeared (published), in the 18th century, there is a probability that the nursery rhyme refers to the Ming empire and its reliance on the Great Wall. And that it was initially a diplomatic secret missive.

Modelling this in a Google or semantic search framework actually yields good results. Once you have eliminated "irrational" explanations such as the YouTube above, you will find out that the "wall" and "great fall" rhymes can yield a "great wall" search result and that the published date, if we put it at 100 years or 50 years after the event, as in most classified government documents, more or less matches the fall of the Min empire.

Other texts are for example "Finnegan's Wake" by James Joyce... This is an excerpt from the beginning:

riverrun, past Eve and Adam's, from swerve of shore to bend of bay, brings us by a commodius vicus of recirculation back to Howth Castle and Environs. Sir Tristram, violer d'amores, fr'over the short sea, had passencore rearrived from North Armorica [...]

So now, how does an NLP/AI framework deduce that the text is 1) rejecting humanism, rationalism and the renaissance (assuming it is) 2) rebuilding a pseudo language spoken by the English after the fall of the Roman empire 3) is a significant work in the 19th century-early 20th century "language crisis" movance... Is there irony?

Consensus in literary interpretation is sometimes important, assuming people talk.

Dear Philon Nguyen!

It is very beautiful:

"...when comparing it to human intelligence is to be able to weight "What is said" vs "What is not said". + my very strange old question: someone suddenly remembered a forgotten fact. How does he know that it's right ???

Joseph

Peter :"I really don't know what unconventional mathematics could be."

I basically agree with you, except for the following---which may turn out to be very important---scenario that you missed.

What we have never considered in mathematics (and science in general) is the important possibility of non-numeric,'structural', forms of data representation, where numbers are replaced by some 'non-trivial' structured entities. The resulting structural math will have very little in common with the known math.

Of course, to be meaningful and useful such a form of representation must reveal completely new, "structural", side of reality and must therefore rely on the fundamentally new "measurement" tools.

For an example of such representational formalism see a popular introduction at

http://www.cs.unb.ca/~goldfarb/BOOK.pdf

and some other my papers here at RG.

Dear Peter Breuer,

What is mathematics: (from Internet)

1.The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.

2. A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement

3. The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Encyclopaedia Britannica

4. A mathematician is a blind man in a dark room looking for a black cat which isn't there. Charles Darwin

5. Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined

Peter,

It appears you are not reading answers carefully, ;--)

I said, that "to be meaningful and useful such a form of representation must reveal completely new, "structural", side of reality and must therefore rely on the fundamentally new "measurement" tools."

Please note a "completely new, 'structural', side of reality", and please point to any such new forms of *data representation* in mathematics. (I'm surprised you didn't ask me anything about "data representation".).

Peter: "The problem is that I don't really have a clue what the assembly of words you've put together above actually _means_. What is "reality"? What is a "structural" aspect of it, once we know what you mean by it? What, once the other stuff is settled, is "new" about the view you propose (but have not described or defined)? What is a "'measurement' tool"? Why is "measurement" in quotes?"

That's a good start, but I had to ask for it. ;--)

However, the surprising thing is that somehow you are now asking for all kinds of clarifications *except* for my key point: "non-numeric,'structural', forms of data representation, where numbers are replaced by some 'non-trivial' structured entities".

As far as I know, so far, mathematics has not been explicitly concerned with various forms of data representation. Isn't that the reason you ignored it?

For some aspects of mental phenomena, the simple use of mathematics is adequate, but when it comes to the social activity of humans supported by the mental, it thoroughly falls short of goal mentioned.

Peter,

It appears that after taking all those "non-numeric" courses, somehow, you missed the important historical perspective. In particular, most of this non-numeric stuff, which appeared during the last century, is a very thin (often not tasty at all and quite temporary) icing on the very thick cake that has been baked for several thousand years. By the way, the production of this ephemeral icing became possible only because the society allowed the number of mathematicians (and other scientists) to grow incredibly. However, the main problem with this growth is that, in the absence of great visionaries---which we desperately need in the wake of the present unprecedented transitional period---it turned out that the same growth serve as a great obstacle to the impending radical changes. For example, you yourself are, in fact, suggesting that "we have already seen just about everything 'non-numeric' we can muster, so we should not expect any surprises here".

Here is a relevant point. As you might know, John von Neumann was one of the very few mathematicians of the last century with the exceptional sense of abstract structures in mathematics, and here is what he had to say in the last two paragraphs of "The Mathematician" (1947) :

"As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired from ideas coming from 'reality', it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l'art pour l'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much 'abstract' inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this would be too technical.

In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas. I am convinced that this is a necessary condition to conserve the freshness and the vitality of the subject, and that this will remain so in the future."

But, despite your impression, we had *very* little of non-ephemeral "non-numeric" stuff. And the structural data representation I was talking about has nothing to do with the latter.

So what I suggest is that, among the non-numeric representations, we should take seriously only those developments that are directly motivated by some *fundamental* applied considerations coming from *natural sciences*. This is a much 'safer' and more reliable strategy.

Dear Peter,

I tried to get you involved in the process of gradual clarification of the meaning of structural representation, since there is absolutely no known to you direction I can point to.

However, since you insist on a direct clarification, I would suggest the long introduction in http://www.cs.unb.ca/~goldfarb/BOOK.pdf

and

one of the shorter (last) introductions in http://www.cs.unb.ca/~goldfarb/FQXi_5.pdf .

There is a formal exposition in: http://www.cs.unb.ca/~goldfarb/ETSbook/ETS6.pdf .

Dear Lev Goldfarb,

Thank you for very interesting articles!

Although this topic does not belong to the sphere of my studies

I was attracted to the philosophical problem of knowledge.

Joseph,

You are more than welcome!

Peter,

First of all, it is hard to explain the basic idea of the proposed ETS formalism because of the promiscuity of such terms as "event", "structural representation", etc. I suggest: *do not* rely on the conventional meanings of such terms.

Also, don't think of "events" as operating in space: they are informational (abstract) specifications that can be *instantiated* spatially. Think of an event as a new formal tool for specifying structural transformation (as illustrated in Fig. 4 of http://www.cs.unb.ca/~goldfarb/FQXi_5.pdf ).

A helpful *actual* process to keep in mind is the process of embryo development: each new 'event' transforms the embryo *structurally*.

A very simple, illustrative, example is in the endnote 4 of the same non-technical paper. A "struct" is a stream of interconnected events. Struct is supposed to capture the corresponding *formative* object structure.

Among my papers here at RG after the year 2000, you will find some more involved illustrative examples.

I think that the question in this thread must be split into two different sentences.

1) Is it possible to describe the brain structure by means of conventional mathematics?

2) If the preceding question has an affirmative answer, then is it possible to imitate brain structure in computational science?

The first question has a simple answer. Since maths are extensible, if it is not possible, the question is if somebody will be able to extend maths in order to be possible. The answer is a matter of future human history.

The second question depends on what are meant by computational science. If it is bounded to those machines that need to be programmed, the answer is not. Humans are able to learn without being programmed. Of course, there are some native structure in human mind, like conditioned reflexes. The true question is to know what is the minimal structure S that it is required in order to learn and extend his own capability. Indeed, conditioned reflexes lie in S.

Some related questions are analyzed in http://scik.org/index.php/esl/article/view/785

In any case, answering this question is only possible if the human brain is able to know his own structure. To be comparable with a human brain, any artificial machine must possess the same capability.

Dear Juan-Esterban,

If the brain structure is "the structure" in our sense ? My be

it is a very dynamic thing ? For each solution we have a new "computer program".

I thing that we cannot transfer our primitive mechanical solutions to the brain actions.

Dear Joseph,

Since we are speaking about maths possibilities, "the brain structure" must be understood as the "algebraic structure of the human brain working way", absolutely disregarding any computer concept.

In addition, you have said: "I thing that we cannot transfer our primitive mechanical solutions to the brain actions."

I agree with you. Brain machinery is far from mechanics laws. The most elementary brain action consists of conditioned reflexes, which are the basis for memory.

Conditioned reflexes depend on repetition frequences, by no means they depend on forces or physical fields.

This fact is known by any student. For instance, to set up associations of the form "country-capital", one repeat again and again both words, the name of the country followed by the corresponding capital name. Abstractions have a more complex algebraic structure requiring the use of logics.

In any case, I have not mentioned mechanics in my message. I cannot understand how you have interpreted my message in terms of mechanics; however, conditioned reflexes can be imitated by physical gadgets easily. Even it is possible to generate abstractions, but the complexity of this topic requires a large article, which does not fit into this web page.

Simen: "there is every reason to believe that Peter is very well versed in not relying on the conventional meaning of terms."

First of all, I meant not just "conventional", I meant all formal (or even informal) meanings of the terms. Model theory has hardly anything to do with what I'm talking about.

Simen, it would be more productive if you would try to understand a bit what I'm talking about. ;--)

20th century mathematics has also nothing to do with this. Please read very short 3rd chapter (which I just added to http://www.cs.unb.ca/~goldfarb/BOOK.pdf ).

Dear Juan-Esteban,

In the context of this discussion, the brain's neurophysiological structure is not of interest. The relevant architecture of the brain evolved to support some informational processes and is in many ways accidental.

What should be of interest to us is the central information process itself (for example, pattern recognition / induction), which I believe could not have emerged independently during the biological evolution.

Dear Lev,

You have written: "the brain's neurophysiological structure is not of interest".

I have read again and again my two messages and I cannot find any mention to the "brain's neurophysiological structure" in them. My only interest is the algebraic structure of intelligent system behavior. For instance, the algebraic category of automata and its subcategory of Turing machines, correspond to some mind processes.

The question is whether intelligent systems can be formalized by means of mathematical language and mathematical concepts.

In addition you have said "I believe could not have emerged independently during the biological evolution." I agree with you, and I cannot understand why you use this claim seemingly against my message. To see how far I agree with this topic, I must inform you that I consider that the every evolution process is intelligent. Obviously, I assign a wide meaning to the term "intelligent".

Take into account that when a human tries to solve some problem, usually, he builds several solutions that occur in his mind, partly by what he knows and partly at random. By testing all obtained constructions, he reject those that are not solutions for the problem until he finds the adequate one. Is not this an evolution process?

In evolution, heredity is what the evolving being knows, but mutations are the random part of the process. Notice that both processes are quite similar.

The only difference between an evolutive process and any intelligent one is that the first process occurs in the external world, while the second occurs inside a human mind, that is to say, it is a "Gedanken" experiment or mental process.

Kind regards

Juan-Esteban

P.S.

Notice that to comment your message, I have quoted your claims. This is a good practice in order to avoid reasoning upon what it is not said.

Juan-Esteban,

OK, wrt to my my earlier post, since you asked for the relevant quote from your post, were you talk of "brain structure" here it is:

"1) Is it possible to describe the brain structure by means of conventional mathematics?"

It was natural for me to assume that you are talking about the "brain's neurophysiological structure", isn't it?

Also, I believe the "algebraic structure of intelligent system behavior" is not relevant, for the reasons I explained in http://www.cs.unb.ca/~goldfarb/BOOK.pdf .

Basically, before addressing any "algebra" we need to address the issue of structural representation, since we might be dealing not with any "algebra" in sense we are used to but with something completely unfamiliar, as von Neumann was suggesting.

Dear Lev,

In a previous message I have explained that by brain structure, in a mathematical context, I am talking about "the algebraic structure of the brain working way , which is far from the physical and neurological brain structure. I have explained it to avoid to be misinterpreted.

Thus, you are not being fair quoting a phrase of mine disregarding the explanation I have written before.

Dear Juan-Esteban,

Please note that I relied on your "brain", in part, because there is no "algebraic structure of the brain". In fact, this is not a helpful, if not confusing, way to approach the issue.

Dear Lev,

Once again you are not being fair. I have never mentioned " algebraic structure of the brain". What i have written is "algebraic structure of the BRAIN WORKING WAY". The brain working way can be formalized in terms of algebraic structures. For instance, the basic brain procedures consist of conditioned reflexes, which can be described by means of asociation algebra. Likewise, categories of automata and Turing machines describe other procedures of the brain working way. At least, my brain is able to handle these structures, besides, I am also able to write the word "apologize" when it is required. Recall, that mathematical constructions belong either to algebraic categories or topological ones together with their particular cases like monads, topoi and quasitopoi.

Juan-Esteban,

I'm sorry, but to me "algebraic structure of the BRAIN WORKING WAY" is as confusing as the one I mentioned.

How can we understand the phrase "algebraic structure of the BRAIN WORKING WAY", algebraic structure of what?.

Why "algebraic structure" instead of simply "structure"? Historically, the idea of "algebra" appeared in the context of various structures emerging at first as axiomatic 'tools' to study various concrete and abstract spaces, including vector spaces. So in our context there are *no* spaces or structures of any *known* kind to apply to them the same historical 'logic'. Moreover, I claim that we will not be able to apply the same 'logic', since we are dealing with completely different kind of entities, not related to the familiar set-theoretical constructions.

Dear Lev,

Of course, human brain works driving stomach and other organs, but when we talk about brain in this context, mainly, we are referring to those actions that are related with thought.

Thought consists in sentences, which are word sequences having a meaning. Accordingly, the corresponding algebraic structure consists of an endofunctor F:Set →Set sending each set into a member-sequence collection. Thus, any natural transformation µ:X→ F(X) defines a co-monadic co-algebra and the question is reduced to find out the adequate axioms that such a co-algebra must satisfy in order to give rise to sequences in a similar way as any human brain does.

Now, every sequence can also be built by an association system assigning a successor a_(n+1) to each sequence element s_n under some state set. The most natural way of defining such an association system is by means of a monad associated to another endofunctor M:X → X×∑ together with the corresponding Kleisli category, where ∑ stands for the corresponding state set. An example of this algebraic structure is the well known Haskell programming language based on monads. The algebraic structure, that any system that works in a similar way as human brain must satisfy, consists of a pair of endofunctors like F and M. However, the most foundational feature of human brain is the capability of extending its own skills. This circumstance requires other functors extending both F and M and depending on information sets obtained from the real world. These extensions must be of heuristic nature, for not every question can be answered by any human being. In addition, human brains do not work with blind phrases, but sentences. Recall that a sentence is a phrase having a meaning. Accordingly, another association algebra is required, namely, an axiom set by means of which a meaning is assigned to each phrase. This association is also state dependent; therefore must be structured by means of an endofunctor defining a monad. Thus, in this abstract level, the structure of those systems working as a human brain can be described by means of monadic and co-monadic algebras.

Notice that the structure I have defined is wider than the automaton one, in particular, Turing machines. The working way of a Turing machine consists of tape-configuration sequences that are nothing but symbol sequences. Accordingly they can be described by an endofunctor as the above defined F.

Juan-Esteban & Simeon,

You missed my point quoted (but completely misunderstood) by Simeon. (It has nothing to do with ordinal vs cardinal.)

My point is this. Look at the struct proposed by us. This struct is a *temporal* generalization of natural number. But one has to understand it as a *structural representation* of an evolving process/object: as the process/object continues to evolve, struct is growing. Set-theoretic logic is not applicable here, since the basic entities are not points but evolving structs.

Dear Lev,

My messages have no relation with your book. Those of my messages that I have adressed to you, have been written to correct the biased quotations that you have written about a post of mine.

Nevertheless, I have taken a glance at your book and I can see that it contains no proofs for any of your claims. This fact makes me suppose that it is of philosophical nature, because maths are built upon axiomatic method, consisting of axioms and theorems, by no means math are obtained by opinions, illumination, intuition or personal preferences. Since my research field are maths, I have no word to say about your book. In any case, my opinions are very poor, mainly, because I am not fond to opinions. My hobby are proofs.

Regards.

Juan-Esteban

Juan-Esteban,

As I mentioned above to Peter, for proofs see

http://www.cs.unb.ca/~goldfarb/ETSbook/ETS6.pdf .

But since you are into "proofs", I am surprised you are interested in the topic of intelligent information processing which is not yet ready for the "proofs only" game. ;--)

Dear Lev,

Of course, there are very interesting things apart from maths, but my interest is focused on proofs and logic. i'm living in a free country where everybody can prefer what he likes. Do not worry if my interest does not coincides with yours. Variety is a good attribute of mankind.

Dear Juan-Esteban,

You again misunderstood me. What I was saying is that intelligent information processing (as von Neumann concluded already in 1956) cannot be approached based on the math formalisms as we know them today.

There is absolutely nothing wrong with hunting for nice proofs and theorems (I did it when I was much younger), but this activity has nothing to do with trying to develop *radically* new formalisms for addressing the needs of intelligent information processing. And the latter have nothing to do with various combinations of the existing formalisms.

Dear Lev,

I have understood you perfectly. This is why I know that I am not interested in your opinions. I am no able to appraise any opinion or proof-free claims. I apologize, but need not change my preferences. Please, do not insist.

It's really funny!

I am meeting here at RG many people who think that model development is "philosophy".

You are right!!! it is very funny!!!!!!

Simen,

I just noticed that I haven't answered your points above (Sept. 19), mainly because I was answering Juan-Esteban.;--)

1. "First, the distinction between ordinal number and cardinal numbers is very clear in mathematics, despite your claim that it has been "blurred" "

This distinction emerges only when the corresponding *additional* structures (e.g. ordering) are introduced. The number itself, e.g. 2, doesn't have that information (think of the abstract entity "2").

2. "Now, if you somehow feel that our perception of physical reality is inadequate you should probably reform physics or some other applied field, not mathematics. Mathematics in this context is merely a tool used by physicists to describe their understanding of physical reality. "

You forget that without the mathematics as we know it there would be no physics.

3. "Mathematics may perfectly well be used to model the passage of time, orderly events, temporal dependence, what not. There is no need to put time intrinsically into the natural numbers, or be aware of the temporal roots of natural numbers for this to succeed. "

If you were right, we wouldn't have a big debate about the nature and the role of time among physicists.

You are missing a key point I'm advocating: time will only enter "mathematics" when we replace its basic entities (such as numbers) by others embodying some kind of 'irreversibility'. Today, in mathematics, we don't have such *basic* entities. Number "2" tells us very little about the passage or nature of time.

@Lev Goldfarb: First of all I would like to declare that the renewal of mathematics is expected through Computer Science, AI, and Biology. I just have a look at one of your papers, and I think what you call "struc" is very similar with braids. I suggest to have a look at Section 2.4: Braided Monoidal Categories. of the following paper, known as Rosetta Stome.

@Costas, as I often have had to point out, the resemblance seems to be only superficial. Keep in mind that, in the case of our ETS, there are no spatial, including topological, considerations involved. They come in only at a later stage, during the spatial *instantiation* of structs.

@ Lev: OK. Lev you know much better than me. What I thing I saw is that many of the examples are done using Category Theory. I may try to collect such examples, and then we can talk agian.

Sure, Costas! Thank you! (But check the concept of event in ETS.)