I haven’t written
anything meaty for a while, and I’m worried I might lose my touch. Besides, I
feel the need to stimulate my brain and, hopefully, yours in the process.

Just before Christmas,
I read an excellent book by Noson S. Yanofsky, titled:

*The Outer Limits of Reason; What Science, Mathematics, and Logic CANNOT Tell Us*. Yanofsky is Professor in the Department of Computer and Information Science at Brooklyn College and The Graduate Center of the City of University of New York. He is also co-author of*Quantum Computing for Computer Scientists*(which I haven’t read).
Yanofsky’s book (the
one I read) covers a range of topics, including classical and quantum physics,
chaos theory, determinism, free will, Godel’s Incompleteness Theorem, the P-NP
problem, the anthropic principle and a whole lot more. The point is that he is
well versed in all these areas, yet he’s very easy to read. His fundamental
point, delineated in the title, is that it is impossible for us to know
everything. And there will always be more that we don’t know compared to what
we do know. Anyone with a remote interest in epistemology should read this
book. He really does explain the limits of our knowledge, both theoretically
and practically. At the end of each section he gives a synopsis of ‘further
reading’, not just a list. I found the book so compelling, I even read all the
‘Notes’ in the appendix (something I rarely do).

Along the way, he
explains things like countable infinities and uncountable infinities and why it
is important to make the distinction. He also explains the difference between
computing problems that are simply incomputable and computing problems that are
computable but would take more time than the Universe allows, even if the
computer was a quantum computer.

He discusses, in
depth, philosophical issues like the limitations of mathematical Platonism, and
provides compelling arguments that the mathematics we use to describe physical
phenomena invariably have limitations that the physical phenomena don’t. In
other words, no mathematical equation, no matter its efficacy, can cover all
physical contingencies. The physical world is invariably more complex than the
mathematics we use to interpret it, and a lot of the mathematical tools we use
deal with large scale averages rather than individual entities – like the
universal gas equation versus individual molecules.

He points out that
there is no ‘fire in the equations’ (as does Lee Smolin in

*Time Reborn*, which I’ve also read recently) meaning mathematics can describe physical phenomena but can’t create them. My own view is that mathematics is a code that only an intelligence like ours can uncover. As anyone who reads my blog knows, I believe mathematics is largely discovered, not invented. Marcus du Sautoy presented a TV programme called The Code, which exemplifies this view. But this code is somehow intrinsic in nature in that the Universe obeys laws and the laws not only require mathematics to quantify them but, without mathematics, we would not know their existence except, possibly, at a very rudimentary and uninformed level.
Yanofsky discusses
Eugene Wigner’s famous declaration concerning ‘The Unreasonable Effectiveness
of Mathematics’ and concludes that it arises from the fact that we use mathematics
to probe the physical world, and that, in fact, leaving physics aside, there is
a ‘paucity of mathematics in general science’. But in the next paragraph, Yanofsky says this:

*The answers to Wigner’s unreasonable effectiveness leads to much deeper questions. Rather than asking why the laws of physics follow mathematics, ask why there are any laws at all.*

In the same vein,
Yanofsky gives a personal anecdote of a student asking him why complex numbers
work for quantum mechanics. He answers that

*‘…the universe does not function using complex numbers, Newton’s formula, or any other law of nature. Rather, the universe works the way it does. It is humans who use the tools they have to understand the world.’*And this is completely true as far as it goes, yet I would say that complex numbers are part of ‘the code’ required to understand one of the deepest and fundamental mysteries of the Universe.
Yanofsky’s fundamental
question, quoted above, ‘why are there any laws at all?’ leads him to discuss
the very structure of the universe, the emergence of life and, finally, our
place in it. In fact he lists this as 3 questions:

*1: Why is there any structure at all in the universe?*

*2: Why is the structure that exists capable of sustaining life?*

*3: Why did this life-sustaining structure generate a creature with enough intelligence to understand the structure?*

I’ve long maintained
that the last question represents the universe’s greatest enigma. There is
something analogous here between us as individuals and the cosmos itself. We
are each an organism with a brain that creates something we call consciousness
that allows us to reflect on ourselves, individually. And the Universe created,
via an extraordinary convoluted process, the ability to reflect on itself, its
origins and its possible meaning.

Not surprisingly,
Yanofsky doesn’t give any religious answers to this but, instead, seems to draw
heavily on Paul Davies (whom he acknowledges generously at the end of the
chapter) in providing various possible answers to these questions, including
John Wheeler’s controversial thesis that the universe, via a cosmic scale
quantum loop, has this particular life and intelligence generating structure
simply because we’re in it. I’ve discussed these issues before, without coming
to any definitive conclusion, so I won’t pursue them any further here.

In his notes on this chapter, Yanofsky makes this point:

*Perhaps we can say that the universe is against having intelligent life and that the chances of having intelligent life are, say, 0.0000001 percent. We, therefore, only see intelligent life in 0.0000001 percent of the universe.*

This reminds me of
John Barrow’s point, in one of his many books, that the reason the universe is
so old, and so big, is because that’s how long it takes to create complex life,
and, because the universe is uniformly expanding, age and size are
commensurate.

So Yanofsky’s is a
deep and informative book on many levels, putting in perspective not only our
place in the universe but the infinite knowledge we will never know. Towards
the end he provides a table that summarises the points he delineates throughout
the book in detail:

*Solvable computer problems Unsolvable computer problems*

*Describable phenomena Indescribable phenomena*

*Algebraic numbers Transcendent numbers*

*(Provable) mathematical statements Mathematical facts*

Finally, he makes the
point that, in our everyday lives, we make decisions based primarily on
emotions not reason. We seemed to have transcended our biological and
evolutionary requirements when we turned to mathematics and logic to comprehend
phenomena hidden from our senses and attempted to understand the origin and
structure of the universe itself.