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Mathematical Structures of Natural Intelligence

Yair Neuman
Publication Date: 
Number of Pages: 
Mathematics in Mind
[Reviewed by
David S. Mazel
, on

This slim volume is a theoretical view for a formal approach to the knowledge of objects. The author explores Category Theory as a way to classify objects and understand the relation between objects. This approach seems to me to be one method to understand common words and their relation between one another. While the text is quite theoretical I will only outline some of the ideas here.

For example, let’s say you want to look at a jump rope and an electric cord. You might produce the following diagram to describe these two objects.

Figure 1: Learning about common objects: A jump rope and an electrical cord.

In the middle row are the objects and a description of them; both are like ropes. The objects share similarities (the middle column) but they also have differences (first and last columns). This type of diagram gives you a way to think and classify these objects. It is what I would expect to see in a program for artificial intelligence needed to identify a jump rope or electric cord and still be able to distinguish the two. This is what the first part of this book discusses.

But what about a change to an object itself. Let’s say we have an apple and there are successive bites taken from the apple. Can we discuss the change to the apple as it is consumed and transformed for a whole apple to a core? Yes, we can. Figure 2 shows the transformation and Figure 3 shows a structure of the change as a groupoid.


Figure 2: Consumption of an apple. (Fig. 6.1 of the text)



Figure 3: First three steps of the apple consumption as a groupoid. (Figure 6.3 of the text)

Again, this is an approach to representing knowledge of the apple. Above we discussed a way to think of apples as fruit and possibly relate it to a pear. Here the author shows us how to think of transformations of an apple.

Neuman provides further work to classification where, for example, he shows the reader how to decompose an object into pieces. One example he uses is to decompose a wasp into wings and body. Thus we begin to have a formal approach to resolve an object into its constitutive components.

The gist of this book is a mathematical approach to understand and work with objects. The author provides us with a method to use mathematical concepts I had not previously seen.

At the end of the book, the author tells us the “book’s main benefit may be summarized as a challenging experience for the intellectual.” While I am not an intellectual, the book is a challenging experience with its mix of mathematical formalism applied to objects and knowledge.

Further the author cites three other benefits to the book:

  1. A “better understanding of the way natural intelligence identifies and forms orders existing in the world.”

    I would say the book does this partially. While the author provides clear examples, I am not sure this benefit is completely satisfied.
  2. The book provides “theorizations” with “descriptions of natural intelligence’s various expressions.”

    I cannot say I saw this accomplished.
  3. “The reader may find the theoretical Lego blocks provided herein relevant for modeling processes in his or her own domain…”

    On this point the author succeeds quite well. Neuman gives the reader a mathematical foundation to understand how to model knowledge and relationships. While he does not provide much in the way of practical advice, nor does he cite others who could do so, this book is a start and could be of help to the practitioner.

Overall, this book is a short theoretical approach to organize, relate, and structure knowledge. It is purely theoretical. If you seek something beyond theory, you will need to look elsewhere. 

David S. Mazel is a practicing engineer in Washington, DC. He welcomes your thoughts and feedback. He can be reached at mazeld at gmail dot com.

See the table of contents in the publisher's webpage.