Mathematics as "Virtual Physics"

[merjet]

http://cogprints.org/385/0/Mathem.html

This is rather long -- 35 pages when I copied it to a Word document.

I found it very interesting. Topics include the nature of proof, logic, philosophy of math, history of math, pure and applied math compared.

Enjoy!

[tjohnson]

In other words, mathematical objects are neither invented nor discovered : they are a cultural production, they are created as part of human activity, the same way as a price is created in a commercial transaction, or they are conceived within a brain, i.e. within a human body in the same way as a child is conceived. A new theorem is very much like a neologism, a new word being coined, i.e. it is a way of « chunking » some existing conceptual reality, a new way for assigning a nickname to a complex of concepts, a new way for a stylization of the world. In the fourteenth century William of Ockham wondered where universal concepts do come from. The answer is that they do not come from any foreign location, they are created within language through human industry. Any time a new concept - a new theorem - is created, the world is different from then on, that is, if the memory of it survives, if its transmission gets accomplished. Culture is vulnerable to lack of transmission in the same way as a species depends on transmission of its genetic pool : culture requires « depositories » in the shape of memories contained within brains, it requires supports, in the same way as a radio transmitter is nothing without receivers.

Yes, I think Korzybski would agree.

It makes mathematics a linguistic scheme which

embodies the possibility of perfection, and which, no doubt, satisfies semantically,

at each epoch, the great majority of properly informed individual Smiths and

Browns. There is nothing absolute about it, as all mathematics is ultimately a

product of the human nervous system, the best product produced at each stage of

our development. The fact that mathematics establishes such linguistic relational

patterns without specific content, accounts for the generality of mathematics in

applications....

...Theories are relational or structural verbal schemes, built by a process of high

abstractions from many lower abstractions, which are produced not only by

ourselves but by others (time-binding). Theories, therefore, represent the shortest,

simplest structural summaries and generalizations, or the highest abstractions from

individual experience and through symbolism of racial past experiences. Theories

are mostly not an individual, but a collective, product. They follow a more subtle

but inevitable semantic survival trend, like all life. Human races and epochs which

have not revised or advanced their theories have either perished, or are perishing.

[BaalChatzaf]

http://cogprints.org/385/0/Mathem.html

This is rather long -- 35 pages when I copied it to a Word document.

I found it very interesting. Topics include the nature of proof, logic, philosophy of math, history of math, pure and applied math compared.

Enjoy!

You are aware that mathematics, done abstractly, has zero empirical content.

On the other hand, physics is empirical right down to the ground floor. Mathematics is not the content of physics, it is the best way to express quantitative physical laws and hypotheses. Mathematics is one of the items in the tool kit of physics.

So mathematics, qua mathematics is not any kind of physics, real or virtual.

Ba'al Chatzaf

Edited November 7, 2007 by BaalChatzaf

[merjet]

You are aware that mathematics, done abstractly, has zero empirical content.

I don't deal in floating abstractions.

Do you agree with David Hilbert's view of mathematics called Formalism? See particularly the first paragraph in the Formalism section here.

[BaalChatzaf]

Do you agree with David Hilbert's view of mathematics called Formalism? See particularly the first paragraph in the Formalism section here.

By and large, yes. Mathematics become serious (i.e. a non-game) when the objects of the formalism are mapped or associated to objects or processes that can be observed. For example measurements of various sorts. At that point the mathematics is no longer abstract, but concretized modulo some interpretation (i.e. a mapping into the real world). That is, more or less, how mathematics is transformed into physics. This the elements of an abstract Hilbert Space are interpreted as quantum states and observations as Hermitean operations on a Hilbert Space. The nice thing about serious (applied) mathematics is that being serious, it looses non of its abstract power and generality.

Eugene Wigner was very much taken with how readily abstract mathematics, never developed for applications initially, can be put in service of applications.

Ba'al Chatzaf

Edited November 7, 2007 by BaalChatzaf

[BaalChatzaf]

You are aware that mathematics, done abstractly, has zero empirical content.

I don't deal in floating abstractions.

Do you agree with David Hilbert's view of mathematics called Formalism? See particularly the first paragraph in the Formalism section here.

Aristotle came rather close to a correct view on the matter, in spite of the fact that he (and other Greek philosophers) did not know all that much mathematics).

Read Book II Chap 2 of -Physics-.

Aristotle hits pretty near the mark in differentiating what mathematicians do and what students of nature do.

Ba'al Chatzaf

[merjet]

Aristotle came rather close to a correct view on the matter, in spite of the fact that he (and other Greek philosophers) did not know all that much mathematics).

Read Book II Chap 2 of -Physics-.

Aristotle hits pretty near the mark in differentiating what mathematicians do and what students of nature do.

Agreed. Where would he best fit in the schools described in the Wikipedia article? I don't believe it would be Formalism, at least not that of David Hilbert's later view that most of math is meaningless. Constructivism probably.

Eugene Wigner was very much taken with how readily abstract mathematics, never developed for applications initially, can be put in service of applications.

I believe mathematics that was developed for applications initially outweighs what wasn't.

Btw, Aristotle said optics, harmonics, and astronomy were branches of mathematics. (from a modern perspective)