I will argue that it is a category mistake to regard mathematics as “physically real”. I will quote D. E. Littlewood from his remarkable book “The Skeleton Key of Mathematics”:

“A trained sheepdog may perceive the significance of two, three or five sheep, and may know that two sheep and three sheep make five sheep. But very likely the knowledge would tell him nothing concerning, say horses. A child learns that two fingers and three fingers make five fingers, that two beads and three beads make five beads. The irrelevance of the fingers or the beads or the exact nature of the things that are counted becomes evident, and by the process of abstraction the universal truth that 2+3=5 becomes evident.”

I can just hear someone pouncing on Littlewood’s phrase “universal truth”, but that is not the reason I quote this paragraph. I quote it to remind that the essence of mathematics is never in the particular way it is represented, but in the concept that it brings forth, and the unification of particulars that it embodies.

One might hypothesize that any mathematical system will find natural realizations. This is not the same as saying that the mathematics itself is realized. The point of an abstraction is that it is not, as an abstraction, realized. The set { { }, { { } } } has 2 elements, but it is not the number 2. The number 2 is nowhere “in the world”.

I argue that one should understand from the outset that mathematics is distinct from the physical. Then it is possible to get on with the remarkable task of finding how mathematics fits with the physical, from the fact that we can represent numbers by rows of marks | , ||, |||, ||||, |||||, ||||||, …

(and note that whenever you do something concrete like this it only works for a while and then gets away from the abstraction living on as clear as ever, while the marks get hard to organize and count) to the intricate relationships of the representations of the symmetric groups with particle physics (bringing us back ’round to Littlewood and the Littlewood Richardson rule that appears to be the right abstraction behind elementary particle interactions).

Search for the right conceptual match between mathematics and phenomena, physical and computational. Understand that mathematics is distinct from its instances. And yet our understanding of the abstractions is utterly dependent on knowing more and more about their instantiations. The domains of the physical and the conceptual are distinct and they are mutually supporting.

It is very hard to take this point of view because we do not usually look carefully enough at our mathematics to distinguish the abstract part from the concrete or formal part. The key is in the seeing of the pattern, not in the mechanical work of the computation. The work of the computation occurs in physicality. The seeing of the pattern, the understanding of its generality occurs in the conceptual domain.

Conceptual and physical domains are interlocked in our understanding.

The figure on the following page shows how C.S. Peirce viewed the relationship between phenomenology and mathematics, among other things. I have always found it very apt.

☞ C.S. Peirce, CP 1.186 (1903)

I will say more about this in the blog, but consider the abstraction of negation used by Peirce and by Spencer-Brown — to wit a mark of containment that I shall denote here by a parenthesis:( ) . This is seen as standing for any containment and it also can be seen as standing for its own particular form of containment. Similarly

|| stands for two and refers to its own twoness. We tend to ground the abstract in iconic patterns like this to form a bridge between the abstract idea and its instantiations.

The part I like the most is “The key is in the seeing of the pattern, not in the mechanical work of the computation. The work of the computation occurs in physicality. The seeing of the pattern, the understanding of its generality occurs in the conceptual domain.”

It reminds me about this quote from Bateson: “… a very wide range of philosophic thinking, going back to Greece, and wriggling through the history of European

thought over the last 2000 years.

In this history, there has been a sort of rough dichotomy and often deep controversy. There has been a violent enmity and bloodshed.

It all starts, I suppose, with the Pythagoreans versus their predecessors, and the argument took the shape of ”Do you ask what it’s made of – earth, fire, water, etc?” Or do you ask ”What is its pattern?” Pythagoras stood for inquiry into pattern rather than inquiry into substance. That controversy has gone through the ages, and the Pythagorean half of it has, until recently, been on the whole submerged half.”

One should modify: ”Do you ask what it’s made of – earth, fire, water, etc?” into

“”Do you ask what it’s made of – earth, fire, water, *bits*, etc?” and reply that it is better to ask ”What is its pattern?”, my guess.

Mathematics is always about pattern. And it gets to be very particular and technical by developing language and formalism for working with pattern. Remarkable and subtle patterns are found in Nature that appear to correspond to abstractions developed far away from the phenomena to which they match. It works both ways!

There is a discussion on G+ here around this post.

Ah yes. I did not directly address the notion of a conscious mathematical structure.

That is because I do not see how a formalism alone can be (or become) conscious.

I start with mathematics understood by a consciousness so that the abstractions are understood or in the process of being understood and the formalisms are being developed. The formalisms alone are nothing, but they can be used by mechanisms such as computers to produce patterns that we may find of great interest! It is fun to imagine that a formal system could come alive in some future computer framework,and that it would exhibit understanding. In order to really understand this, it helps to take a very very simple formalism such as a mark denoting a distinction between its inside and its outside and to ask: What are the minimal properties of a system in which the formalism of the mark is embedded that will allow that larger system to be called conscious of the mark and its meaning as an exemplar of distinction? As Spencer-Brown said, “We take the form of distinction for the form.”.