Abstraction vs Eugene Wigner & Edward Frenkel

Thinking depends upon abstraction. Let Isaac Newton observe an apple falling from a tree. The apple and the tree are concrete objects. The observation consists of processes in Newton’s mind. The processes differ from the concrete objects and leave out a wide range of aspects. This is the definition of abstraction: to leave out aspects. Perhaps nature “thinks” by means of the concrete objects, but a mind necessarily must omit details and can only deal with such abstractions. For example, when Newton suddenly is hit by the idea of the universal law of gravity, then this still is an idea in his mind, and not the real gravity that the apple – and he himself – are subjected to.

Newton discovers the universal law of gravity, (c) marcelgotlib.com

Newton discovers the universal law of gravity, (c) marcelgotlib.com

Edward Frenkel’s reference to Eugene Wigner

There is this quote:

“The concepts that Yang and Mills used to describe forces of nature appeared in mathematics earlier because they were natural also within the paradigm of geometry that mathematicians were developing following the inner logic of the subject. This is a great example of what another Nobel Prize-winner, physicist Eugene Wigner, called the “unreasonable effectiveness of mathematics in the natural sciences.” [ref] Though scientists have been exploiting this “effectiveness” for centuries, its roots are still poorly understood. Mathematical truths seem to exist objectively and independently of both the physical world and the human brain. There is no doubt that the links between the world of mathematical ideas, physical reality, and consciousness are profound and need to be further explored. (We will talk more about this in Chapter 18.)” (Edward Frenkel, “Love & Math”, 2013, p 202, my emphasis)

Hopefully you spot the confusion. Frenkel is an abstract thinking mathematician with some experience in science – e.g. with a patent – but apparently without having understood the philosophy of science. This weblog has already discussed some of his views, see here, especially his confusion about mathematics education while he hasn’t studied the empirical science of didactics. It is a chilling horror to hear him lecture about how math should be taught and then see the audience listening in rapture because they think that his mathematical brilliance will certainly also generate truth in this domain.

Eugene Wigner’s error – see the paper below – is to forget that abstraction still is based upon reality. When reality consists of {A, B, C, …, Z} and you abstract from this reality by looking only at A and leaving out {B, C, …., Z} then it should not surprise you that A still applies to reality since it has been taken from there.

Mathematical ideas have a perfection that doesn’t seem to exist in concrete form in reality. A circle is perfectly round in a manner that a machine likely cannot reproduce – and how would we check ? If the universe has limited size then it cannot contain a line, which is infinite in both directions. Both examples however are or depend upon abstractions from reality.

Since mathematics consists of abstractions, we should not be surprised when its concepts don’t fully apply to reality, and neither should we be surprised when some applications do. That is, there is no surprise in terms of philosophy. In practice we can be surprised, but this is only because we are mere human.

Paper on abstraction

This issue on the definition and role of abstraction is developed in more detail in this paper, also in its relevance for mathematics education and our study of mind and brain: An explanation for Wigner’s “Unreasonable effectiveness of mathematics in the natural sciences”, January 9 2015.

A correspondent commented:

“It seems to me that the question that Wigner is asking is “Why is mathematics so much more effective in physics (which is what he means by ‘natural sciences’) than in most other studies?”  Physics textbooks are full of formulas; these comprise a large fraction of what the field is, and have great predictive power. Textbooks on invertebrate biology have few mathematical formulas, and they comprise only a small part of the field. Textbooks on comparative literature mostly have no formulas. So an answer to Wigner’s question would have to say something about what it is about physics _specifically_ that lends itself to mathematimization; merely appealing to the human desire for abstraction doesn’t explain why physics is different from these other fields.
I have no idea what an answer to Wigner’s question could possibly look like. My feeling is that it is better viewed as an expression of wonderment than as an actual question that expects an answer.”
(Comment made anonymous, January 9 2015)

I don’t agree with this comment. In my reading, Wigner really poses the fundamental philosophical question, and not a question about a difference in degree between physics and literature. The philosophical question is about the relation between abstraction and reality. And that question is answered by reminding about the definition of abstraction.

I can agree that physics seems to be more mathematical in degree than literature, i.e. when we adopt the common notions about mathematics. This obviously has to do with measurement. Use a lower arm’s length, call this an “ell“, and proceed from there. Physics only has taken the lead – and thus has also the drawbacks of having a lead (Jan Romein’s law). Literature however also exists in the mind, and thus also depends upon abstractions. Over time these abstractions might be used for a new area of mathematics. Mathematics is the study of patterns. Patterns in literature would only be more complex than those in physics – and still so inaccessible that we call them ‘subjective’.

For example, the patterns in Gotlib’s comic literature about Newton & his apple might be more complex than the patterns in the physics of Newton & his apple, as described by his universal law of gravity. All these remain abstract and differ from the concrete Newton & his apple.

There is no “unreasonable effectiveness” in that Gotlib’s comic makes us smile.


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