The n-Category Café

Wigner said “reality” but wondered why. Pythagorus said so too, but we don’t trust cults as much these days.

Article in the New York Times, and maths education

Terry Tao:

“…surprisingly often the pursuit of one goal can lead to unexpected progress on other goals (cf. Wigner’s ‘unreasonable effectiveness of mathematics’). See also my article on ‘what is good mathematics?’.”

In the maths excerpts I don’t see a direct reference to “independent ways” (and haven’t read the paper to see if the independent bit is important). The one obvious example of a mathematical-ish nature is computability. This was addressed independently by Turing using universal machines, Church using lambda calculus, Post using his rewriting systems before the equivalence of all was noticed. I don’t know the original work well enough to know if (a) they figured out existence of non-computable functions independently and (b) if each approach did it by basically noting you can set things up to use a Cantor-style diagonal argument to show this (so it’s arguably not really independent) or if there’s more to showing non-computability in, say, lambda calculus or Post systems.

Anyway, that’s a vague thought.

Everything depends on what we call an object. The kind of things I see as ‘robust’ are for example some particular objects from geometry. For instance, the projective spaces (or more generaly, the Grassmanians) are robust in the sense they make sense in very different contexts, namely different geometries: it is almost sufficient to make sense of $GL_n$. This makes some other complicated objects robust as well: singular cohomology, de Rham cohomology, K-theory and cobordism are robust in the sense they make sense in very different contexts. This might mean that what we call geometry is quite robust as well: it can be seen in very wide contexts (algebraic geometry, differential geometry, analytic geometry, Faltings almost algebraic geometry, but also tropical geometry, Toën and Vaquié geometry under $Spec(\mathbb{Z})$, Durov geometry over commutative monads, and I don’t speak of their derived versions…

Another example is the theory of categories (say $1$-categories): this is a very robust theory in the sense that the theory of $(\infty,1)$-categories is just some kind of reincarnation of it: most of the basic statements from $1$-category theory extend in a straightforward way to statements in $(\infty,1)$-category theory.

This kind of way of being robust is certainly related to mathematical reality (in the sense of Lautman, say). This is so real (I mean in the practice of the mathematician) that this is also the kind of things we might even hope to axiomatize: for instance, an axiomatic approach to geometries (e.g. J. Lurie already started to do so). This is where the expressive power of category theory becomes wonderful.

This also makes sense about another kind of robustness: there are some very particular objects which spreads everywhere (i.e. in a lot of robust theories as above). For instance, $SL_2$ or the absolute Galois group of $\mathbb{Q}$. But also, sometines, some objects which might look silly at first sight, like the Fibonacci sequence, which we find by counting generating cells in iterated loop spaces.

But I don’t thing we remain attached to robust ideas only because of our poor minds. It is not that there are some part of reality which are simpler: we make our perception of the reality being simple and robust. And before we even try to look at reality, reality is not simple nor complicated. Without an observer, it is just not determined. Simplicity is a matter of choice.

Physicist viewpoint: we often construct examples or counter-examples either by constructing something with a lot of structure or, in an attempt to construct something with no structure, by defining something random. As an example of the last, see Shannon’s use of random codes, the use of random matrix theory to model excited states in nuclei, use of random states in quantum information theory, use of random graphs as expanders, and many, many more. However, usually when you do something random, you again find structure for other reasons: random codes are a way to reach the Shannon capacity bound, random matrix theory has close ties to integrable systems, random states and random graphs also do really well at meeting certain bounds, and so on. So, I think our problem is that we don’t know how to do something _slightly_ unstructured, as it were.

I am fascinated by Wimsatt’s metatheory.

“… But Ockham’s razor (or was it Ockham’s eraser?) has a curiously ambiguous form–an escape clause which can turn it into a safety razor: How do we determine what is necessary? With the right standards, one could remain an Ockhamite while recognizing a world which has the rich multi-layered and interdependent ontology of the tropical rain forest–that is, our world. It is tempting to believe that recognizing such a world view requires adopting lax or sloppy standards–for it has a lot more in it than Ockhamites traditionally would countenance.”

However, this is foundationally vague, as Ocham’s Razor in its traditional form has a plethora of ambiguities and puzzles.

I lean strongly (after writing a 100-page unpublished monograph, out of which two referred papers have so far emerged) towards belief in the axiomatization of Ockham by MDL.

April 2005
8 x 10, 454 pp., 73 illus.
$50.00/£32.95 (CLOTH)

ISBN-10:
0-262-07262-9
ISBN-13:
978-0-262-07262-5
Series
Bradford Books
Neural Information Processing

Advances in Minimum Description Length
Theory and Applications Edited by Peter D. Gr¼nwald, In Jae Myung and Mark A.
Pitt

===========
see also:

MML, MDL, Minimum Encoding Length Inference
http://www.csse.monash.edu.au/~dld/MELI.html

many useful hotlinks
===========

The Minimum Description Length Principle
by Peter D. Gr¼nwald

ISBN13: 9780262072816
ISBN10: 0262072815

The minimum description length (MDL) principle is a powerful method of inductive inference, the basis of statistical modeling, pattern recognition, and machine learning. It holds that the best explanation, given a
limited set of observed data, is the one that permits the greatest compression of the data. MDL methods are particularly well-suited for dealing with model selection, prediction, and estimation problems in situations where the models under consideration can be arbitrarily complex, and overfitting the data is a
serious concern. This extensive, step-by-step introduction to the MDL Principle provides a comprehensive reference (with an emphasis on conceptual issues) that is accessible to graduate
students and researchers in statistics, pattern classification, machine learning, and data mining, to philosophers interested in the foundations of
statistics, and to researchers in other applied sciences that involve model selection, including biology, econometrics, and experimental psychology.

Part I provides a basic introduction to MDL and an overview of the concepts in statistics and information theory needed to understand MDL. Part II treats universal coding, the information-theoretic notion on which MDL is built, and part III gives a formal treatment of MDL theory as a theory of inductive inference based on universal coding. Part IV provides a comprehensive overview of the statistical theory of exponential families with an emphasis on their
information-theoretic properties.

The text includes a number of summaries, paragraphs offering the reader a fast track through the material, and boxes highlighting the most important concepts.

The text includes a number of summaries, paragraphs offering the reader a fast track through the material, and boxes
highlighting the most important concepts.

ISBN:
9780262072816
Author:
Gr¼nwald, Peter D.
Publisher:
Mit Press
Author:
Grunwald, Peter D.
Foreword:
Rissanen, Jorma
Subject:
Mathematical and Statistical Software
Copyright:
2007
Series:
Adaptive Computation and Machine Learning
Publication Date:
May 2007
Binding:
Hardcover
Language:
English
Illustrations:
Y
Pages:
703
Dimensions:
9.14x7.25x1.52 in. 2.74 lbs.

When Wimsatt cites “R. Levins” he is acutely pointing to a foundational argument in mathematical biology, very much on the matter of Occam’s razor as inapplicable.

Richard Levins is discussed thus by the Philosopher of Science Jay Odenbaugh:

The Strategy of “The Strategy of Model Building in Population Biology”
Jay Odenbaugh
Department of Philosophy
Environmental Studies
Lewis and Clark College
Portland, Oregon 97219
jay@lclark.edu

I. Introduction. This essay is an historical exploration of the methodological underpinnings of Richard Levin’s classic essay ‘The Strategy of Model Building in Population Biology’ in which I argue for several theses. First and foremost, his essay constitutes a statement and defense of a more ‘holistic and integrated’ theoretical population biology that grew out of the informal and formal collaborations of Levins, Robert MacArthur, Richard Lewontin, E. O. Wilson, and others. Second, Levins’ essay and the views introduced would be used as a response to the rise of systems ecology in the 1960s against the background of the International Biological Program. Third, the arguments Levins employs are best construed as ‘pragmatic – a point that is sometimes unnoticed by contemporary scientists and philosophers. Finally, I turn to the contemporary and consider the similarities and differences between the limitations of population biology of the 1960s and that of 2005 raising open questions about the applicability of Levins’ analysis.

II. ‘Simple Theorists’ or a Prolegomena to a New Population Biology. In the 1960s, Levins, Richard Lewontin, Robert MacArthur, E. O. Wilson, Leigh Van Valen, and others were interested in integrating different areas of population biology mathematically. Apparently they met on several occasions at MacArthur’s lakeside home in Marlboro, Vermont discussing their own work in population genetics, ecology, biogeography, and ethology and how a ‘simple theory’ might be devised.

In an interview in the early seventies, E. O. Wilson describes the methodological program of the ‘simple theorists’, Biologists like MacArthur and myself, and other scientists at Harvard, Princeton, and the University of Chicago especially, believe in what has come to be called ‘simple theory’, that is, we deliberately try to simplify the natural universe in order to produce mathematical principles. We think this is the most creative way to develop workable theories. We don’t even try to take all the possible factors in a particular situation into account, such as sudden changes of weather or the effects of unusual tides. (Chisholm 1972, 177).

He goes on to compare this program against the competition,

On the one hand, you’ve got the hard ecologists like MacArthur and myself, who, as I’ve explained, believe in simplifying theory as much as possible. You can call us the simple theorists. But in the last five years E. O. Wilson (1994) has written about the meetings that occurred; however, little has been written on the substance of those meetings.

At the time, MacArthur was at Princeton and Lewontin and Levins were at the University of Chicago…. so a group has developed, around people like Paul Ehrlich at Stanford, C. S. Holling at British Columbia in Canada, and Kenneth Watt at Davis, who are also mathematical ecologists, but who believe in complex theory… They say that because ecosystems are so vastly complex, you must be able to take all the various components into account. You really must feed in a lot of the stuff that we simple theorists leave out, like sunsets and tides and temperature variations in winter, and the only way you can do this is with a computer. To them, in other words, the ideal modern ecologist is a computer technologist, who scans the whole environment, feeds all the relevant information into a computer, and uses the computer to simulate problems and make projections into the future. (Chisholm 1972, 181-2)…