FOM: Re: good news: serious physicists believe that intuitionistic logic is important for physics

Peter Apostoli apostoli at
Sat Oct 6 06:49:35 EDT 2001

Dear All,

As far as I know, the first publications by a physicist in the use of topos
theory in quantum gravity were by Chris Isham and Jeremy Butterfield around
1997 to 1998. This was approximately one year after Professor Isham received
a copy of the co-authored book "Parts of the continuum: towards a modern
ontology of science", which at that time had already been reviewed and
accepted for publication by the Poznan Studies in the Phil. of Science and
the Humanities. The book presented a model of semi-constructive (Pi_2) set
theory in the form of CFG (an approximation space placed in type-lowering
correspondence with its own power space). CFG is a topos in the category of
approximation spaces. The book outlined in detail how CFG is the first
successful renormalization of the continuum and suggested that this
quantization of the set theoretic universe is a blue-print for a theory of
quantum gravity. Physicists were the first group to (tacitly) pick up on
this research, but not the first group to originate/develop/publish it. That
honour goes to my co-author, Akira Kanda (who describes himself as a

Prof. Isham describes his research as follows:

>In particular, I have been developing a new approach to quantum gravity
based on a quantum-logic extension of the consistent-histories >version of
quantum theory. This enables the standard ideas of quantum theory to be
extended to situations where there is no normal notion >of time, including
the possibility that a variety of non-metrical aspects of space-time may
also be subject to quantisation. I have become >particularly interested in
the use of generalised set theory ('topoi') in the context of the consistent
histories programme where the internal >logic of a topos seems to play an
important role. I am also planning to use topos ideas in the development of
new models for spacetime; in >particular, there may be important links with
topological quantum field theory, expecially the recent work involving ideas
coming from the >loopspace approach to the Ashtekar programme of canonical
quantisation of gravity.


Peter Apostoli

----- Original Message -----
From: Vladik Kreinovich <vladik at>
To: <fom at>
Sent: Friday, October 05, 2001 7:02 AM
Subject: FOM: good news: serious physicists believe that intuitionistic
logic is important for physics

> Dear Friends,
> For those who like intuitionistic logic and related research, you may want
> read a new popular book by Lee Smolin, one of the world leading
specialists in
> quantum gravity (i.e., quantization of space-time), called "Three rodas to
> Quantum Gravity".
> He strongly believes that this logic - and a more general topos approach -
> of great physical importance to quantum gravity.
> Not only he believes in it, he cites papers by himself and other
> where these logical ideas have been transformed into working physical
> He also emphasizes that the success in physics will eventually lead to a
> successful application of these logics to human decision making, economics
> (Usually, when an idea is used in physics, it is researched over and over
> again, and as a result, in many cases, social sciences have advanced when
> adapted well-developed formalisms from physics).
> Vladik

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