It also means some of those axioms are probably superfluous. It also means your bound by some of the same laws (? maybe the wrong term here?) that computation is (halting problem) etc.
I disagree that his work offers profound new insights or that it has gone underappreciated.
Many mathematicians have explored the connections between computation, logic and formal systems for decades. I also take issue with the assertion that physicists have somehow "forgotten" or failed to appreciate these ideas within computer science. Many physicists are well versed in computational and information-theoretic concepts. The idea that simple rules can give rise to complex emergent behaviors is far from a new revelation.
Furthermore, Wolfram does not give due credit to this rich body of prior work. He presents his perspectives as groundbreaking discoveries, without properly acknowledging similar perspectives developed by others. This is not how science works.
The ability to give credit is something of which Steven Wolfram is in very short supply!
Never mind Wolfram's obvious debt to John Conway, John von Neumann and Alan Turing, among many.
From an interview with Steven Wolfram (FT.com, 11 April 2003. "Lunch with the FT: Stephen Wolfram" by Simon London):
Does he have any heroes, I ask, anyone in science whose work he really
admires?
The question is met with a long pause.
"I think Alan Turing would have discovered a lot of the things I have
discovered if he had lived another 20 years. He was pointed very much in
the same direction."
A hero?
"The more you learn about these people the more difficult it is to idolise
them. Turing was much more of a hacker than I am. He liked putzing around
with things in a way that is somewhat different from my interests."
How about (John) von Neumann, then?
"I'm sure I would have greatly enjoyed meeting him. He was a smart man. But
I would say he fundamentally missed deeper points in a whole series of areas
in his haste to figure things out faster than other people. It was almost
like science as a competitive sport. He skated too quickly over many areas."
Isaac Newton is summarised as "an able man who did interesting work", while
Charles Darwin "had the good feature of living basically a happy, quiet
life" although "the core of what he did is better than some of the
bandwagons he let himself climb on to".
Any living scientist he especially admires? I suggest Stephen Hawking, the
Cambridge astrophysicist, or Benoit Mandelbrot, whose work on fractals seems
adjacent to Wolfram's own interests in complexity.
Again, the reply is careful and considered.
"Hawking I view as a fascinating personal study but as a scientist...well,
I suppose he has done some decent-quality stuff. Benoit Mandelbrot I know
quite well and some of his work is quite important. In fact, I am one of
his bigger promoters. I'm just not sure he has completely internalised
quite how important it is."Thanks for sharing. It is possible Wolfram contributes something to physics by proving how far you can go with (a) constructive mathematics, but it is insane that he seems to miss the equivalences between past work, which is ironic given his entire focus of research.
Computation arises from very simple axiomatic systems, in surprising ways, and that in turn has profound implications for quite a number of things.
But people have been playing with constructive mathematics as a foundation for a long time, but Wolfram doesn't seem to acknowledge that.
I think a lot of us who studied computer science forgot some of the magical epiphanies we had when learning about the nature of computation, so it's always fascinating when Physicists re-derive something in the field.
It's probably similar to how they view it when CS folk wade into discussion on QM and reach enlightenment.
Finally though, I give Wolfram a ton of credit not necessarily for his book, but for actually trying to derive modern physics from (his) version of constructive mathematics.
It's not often you see someone really pushing on something like this in public and willing to discuss their progress in real time.