Dear Yingxu Wang,

I think the real, underlying problem is this: We really do not understand intelligence or, e.g., the semantic aspects of natural language well enough to formulate a promising hypothesis or even a solid model. Thus, the question wether our current state of mathematics is sufficient or not may have to be postponed until we have a better understanding of the subject at hand.

In the long term, I think we will see a form of co-evolution: the more we understand intelligence and natural language, the more we will adapt our mathematical tools to suit or needs.

Cheers,

Jochen

The mathematics of nature seems that it is not appropriate for the mathematics of biological systems and human in particular. The mathematics missing for development of natural intelligence and natural language processing, seems that they are a kind of "soft" mathematics, non-Cantorian mathematics and finally "holistic" mathematics (Category and Topos Theory).  Thus Many-valued logics, fuzzy sets, residuated lattices, etc. are a good start for the mathematics of Biological systems. Since living systems interact strongly, we cannot consider a biological unit as a single point, like in Metric space. Probabilistic matric spaces are more appropriate.

Let me underline the importance of Category Theory. This theory is essentially a kind of social theory. The objects of category interact and furthermore each object is known if we know all ingoing arrows  or all outgoing ones to the object. So if the object is a living entity we do not have to "analyse it by cutting it into pieces, in which case we loose the living entity, but just know the incoming or outgoing arrows. For example, if you want to know "Costas Drossos" you have to know e.g. as much as possible of the ingoing arrows. What are these arrows? The influence of his family, of his schools, of his society, etc.

To day the only interactive mathematics which are applied to Biology is Probability and Statistics. Have a look at the following books.

J.D. Murray Mathematical Biology I. An Introduction

Third Edition

http://www.ift.unesp.br/users/mmenezes/mathbio.pdf

Mathematics for Biologists

http://www-binf.bio.uu.nl/khwjtuss/SystemsBiology/Book/Book.pdf

Look also the following links:

Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better

http://www.plosbiology.org/article/info%3Adoi%2F10.1371%2Fjournal.pbio.0020439

The Society for Mathematical Biology

http://www.smb.org/index.shtml

Dear Yingxu Wang,

I think the real, underlying problem is this: We really do not understand intelligence or, e.g., the semantic aspects of natural language well enough to formulate a promising hypothesis or even a solid model. Thus, the question wether our current state of mathematics is sufficient or not may have to be postponed until we have a better understanding of the subject at hand.

In the long term, I think we will see a form of co-evolution: the more we understand intelligence and natural language, the more we will adapt our mathematical tools to suit or needs.

Cheers,

Jochen

Intelligence of a "system" relates closely to the (degree of) adequacy by which it responds to its "environment". So one might say, for instance, that evolution enhances the intelligence of species in a habitat. On the other hand, intelligent software managing a database involves a handsome representation of the data and secure ways to look up, modify, or add data. Intelligent robots require representing real-world knowledge as a database of facts expressed with an artificial language and an inherent logic.

I don't see any mathematics that is required in all these circumstances. Perhaps logic is the closest to being universally indispensable. In knowledge representation, Boolean algebra is never far away, e.g., in the form of (Tarski's) relation algebra with record-building facilities.

If artificial intelligence is judged to be underdeveloped at the moment, it is probably not mathematics that is lacking, but representations, methods and algorithms. It is basically a problem of computer science and software development (in which mathematics may get involved).

The following is a biased opinion (since I'm working on it) on your question:

Exploiting symmetry in the world and representing them in a Mathematical group theoretical framework (they lend themselves naturally to this) is something that hasn't been exploited much, in my opinion. Natural structures of symmetry can be represented effectively in form of symmetry groups, and the mathematical foundation for this is solid [books on symmetry studies by Weyl, Vianna, etc]. If this mathematical framework can be exploited to represent symmetry present (mostly in man-made, but sometimes natural) world, we will solve a very big representation and generalization problem in AI. It may not be the ultimate solution, but we'll just be one step closer.

I believe that intelligence emerged as an extension of simple control mechanisms ie input -> output->feedback (consequential re-input) -> modified output. This cycle imposed on a state-based internalised representation of  the "real world" generates all of the ingredients required for mathematically modelling - control theory, networks, delay-line style memory (state based) heuristic (adaptive) paths and classification(generalisation using labels (category theory?) ). The recursive nature of feedback provides the mechanisms for heuristic searches or even orderly (symbolically orientated) models which is a "maths".  I suggest that these system, once they are highly refined, can use what we call "logic" a symbolic modelling form which here is a derivative of heuristic processes in which restrictions maintain consistency.  Thus logic is not the best starting point!

If I understand correctly your question, you are referring to perhaps the cognition process and intelligent system. I think neural network has been linked to this type of line of thought.  

In my opinion, human and animals learns from feedback. It is well documented in educational and pedagogic research. If we could model this behaviour we could perhaps enhanced the machine learning. 

In Hyper-heuristics some positive results have suggested the quality of the feedbacks can improve the quality of algorithms solutions found.  In my experience, the better my algorithm fitness evaluation is, the better quality algorithm solution I find.  I have attach one possible interpretation of bio-inspired intelligence  general model.

I hope I understood well your question and answered it.

Patricia 

Conference Paper Plug-and-Play hyper-heuristics: an extended formulation

It can be argued that intelligence requires prediction and all prediction can be encoded as sequence prediction (see e.g. Hutter's work on universal AI). Here is one inspiring paper proving (Gödel-like) the limits of mathematics in the field of sequence prediction.

http://www.vetta.org/documents/IDSIA-12-06-1.pdf