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April 6, 2007

Whatever Happened to the Categories?

Posted by David Corfield

Are we doing our job as broadcasters well? Max Tegmark has a new paper out on the physical universe as an abstract mathematical structure. Not a whiff of categories, let alone nn-categories.

Tegmark has read some of philosophy of science’s ‘structural realism’ literature, but this wouldn’t have pointed him in our direction. Nor would it likely have helped had he looked at philosophy of mathematics’ ‘structuralism’ literature.

Perhaps we’ll have to wait until someone unites quantum field theory and general relativity using a tetracategory before we get noticed.

Posted at April 6, 2007 11:55 AM UTC

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Read the post Tegmark Strikes Again!
Weblog: Science After Sunclipse
Excerpt: Via David Corfield at the n-Category Cafe, here is Max Tegmark’s latest, entitled “The Mathematical Universe”. (Not to be confused with The Mechanical Universe, that Caltech show I grew up watching on PBS and which is now watchable o...
Tracked: April 6, 2007 1:23 PM

Re: Whatever Happened to the Categories?

Tegmark is a cool dude; I could have tried to turn him on to nn-categories at the FQXi get-together in Reykjavik this July, but I’ll be in Delphi talking with you and other friends of Thales! Too much to do…

Posted by: John Baez on April 6, 2007 8:17 PM | Permalink | Reply to this

Re: Whatever Happened to the Categories?

It’s my second paragraph that worries me most. There are some philosophers of mathematics, e.g., McLarty, Marquis, Awodey, who love categories, but they don’t designate themselves ‘structuralist’, a respectable position from the perspective of Anglo-American orthodoxy.

Now there are signs that some philosophers of physics are coming to see the importance of categories, and I rather suspect that this will prove an easier path to achieve wider philosophical attention.

Posted by: David Corfield on April 7, 2007 9:44 AM | Permalink | Reply to this

Re: Whatever Happened to the Categories?

The folks who edited Structural Foundations of Quantum Gravity seemed eager to connect category theory to structural realism — they mention this in the introduction. Despite my qualms with philosophical ‘isms’, I went along for the ride, publishing my paper on quantum quandaries and category theory there.

So, there’s at least a slight chance for fans of ‘structural realism’ to bump into category theory.

But, if there are a bunch of people struggling to understand general concepts of ‘structure’, and how entities attain their individual character by participating in larger structures, and they don’t know category theory yet, somebody should send out a rescue team.

Posted by: John Baez on April 14, 2007 8:34 PM | Permalink | Reply to this

Re: Whatever Happened to the Categories?

Tegmark and I grew up in the same greater area, though he is quite a bit younger than me. I recognized his father from the photos on his webpage, though. He was a math professor when I was an undergraduate, and receives a special thanks in the acknowledgements of the present paper.

Posted by: Thomas Larsson on April 7, 2007 8:53 AM | Permalink | Reply to this

Baiting the hook at OEIS; Re: Whatever Happened to the Categories?

A128818 Examples of integer-encoded mathematical structures in Max Tegmark’s “The Mathematical Universe”.

100, 105, 11120000, 113100120, 11220000110, 11220001110, 1132000012120201



Abstract: “I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters and initial conditions to broader issues like consciousness, parallel universes and Godel incompleteness. I hypothesize that only computable and decidable (in Godel’s sense) structures exist, which alleviates the cosmological measure problem and help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.”


Max Tegmark, The Mathematical Universe, 5 Apr 2007, Table 1, p.3.


“Any mathematical structure can be encoded as a finite string of integers…”

a(1) = 100 which encodes the empty set;

a(2) = 105 which encodes the set of 5 elements;

a(3) = 11120000 which encodes the trivial group C_1;

a(4) = 113100120 which encodes the polygon P_3;

a(5) = 11220000110 which encodes the group C_2;

a(6) = 11220001110 which encodes Boolean algebra;

a(7) = 1132000012120201 which encodes the group C_3.




Jonathan Vos Post (jvospost2(AT), Apr 10 2007

Posted by: Jonathan Vos Post on April 14, 2007 5:59 PM | Permalink | Reply to this

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