I really enjoyed this review of Steven Weinberg's book.

Philip Ball's was a little less positive.

The 1st link comes via Lee's comment on another blog. Full disclosure: I did not read the book.

### the effectiveness of unreasonable math

Scott asked if there is something mysterious about math and, naturally, in the comments "the unreasonable effectiveness of math" came up; i.e. "why the structures that mathematicians found to be important for their own internal reasons, so often turn out to be the same structures that are important for physics".
I wrote the following in response.

One should keep in mind that “the unreasonable effectiveness of math” is possible because the weirdness (e.g. the Banach-Tarski paradox) can be contained and e.g. Goedel’s result does not show up (but it could have, e.g. sums over all possible manifolds in 4d quantum gravity).

But one should also keep in mind that large parts of e.g. the standard model are not even based on well-defined (axiomatic) math, rather a patchwork of “physicist’s math” and the real mystery is why this works.

In other words, the real mystery is "the effectiveness of unreasonable math", an idea which goes back to Vaihinger and his philosophy of "as if".

One should keep in mind that “the unreasonable effectiveness of math” is possible because the weirdness (e.g. the Banach-Tarski paradox) can be contained and e.g. Goedel’s result does not show up (but it could have, e.g. sums over all possible manifolds in 4d quantum gravity).

But one should also keep in mind that large parts of e.g. the standard model are not even based on well-defined (axiomatic) math, rather a patchwork of “physicist’s math” and the real mystery is why this works.

In other words, the real mystery is "the effectiveness of unreasonable math", an idea which goes back to Vaihinger and his philosophy of "as if".

### identity crisis

Obviously, equations are an important feature of mathematics, but there is a problem:

"identity, an objector may urge, cannot be anything at all: two terms plainly are not identical, and one term cannot be, for what is it identical with?" as Bertrand Russell wrote.

This quote is from a blog post of John Baez, which I found fascinating because it exactly reflects my own worries about this question when I was a kid. Of course, then nobody was willing to seriously discuss this with me and it was the first time that I had to conclude that adults are less smart than I initially thought.

A similar problem was caused by my observation that one cannot make valid statements about non-existent things, e.g. the sentence "pink unicorns do not exist" has no meaning, because by definition there is nothing it could be about.

I was convinced for a while that the two problems are related but I am not so sure about this anymore. (*)

I really do regret that the internet did not exist yet when I was a kid, because it would have helped me to refine my worries tremendously (x) and e.g. think about the 'other worlds strategy' earlier (btw I discussed one problem with the idea of other worlds here). On the other hand, I am glad, for obvious reasons, that it did not exist then.

Of course, writing about a non-existent internet is problematic, so instead let me end this blog post with a link to a rather short proof of the famous equation 1+1=2, a proposition which is occasionally useful.

(*) We leave the question whether I really am (or was) the same person as I was (or am?) as a child for another time(!).

However, then I was convinced that if n is a non-existent thing, e.g. the integer n which solves the equation n*n=7, it is not necessarily true that n=n and therefore the concept of mathematical identities is problematic because of non-existent things. But I cannot be sure that I remember this argument from my childhood correctly.

(x) E.g. with a link to Black's symmetric universe.

added later: If you followed this link, then perhaps you already saw also Rene Descartes' argument about human identity: I cannot doubt that I am, however I can doubt that my body (including my brain) exists; it follows that I am not (just) my body.

This conclusion, approximately 350 years before

"identity, an objector may urge, cannot be anything at all: two terms plainly are not identical, and one term cannot be, for what is it identical with?" as Bertrand Russell wrote.

This quote is from a blog post of John Baez, which I found fascinating because it exactly reflects my own worries about this question when I was a kid. Of course, then nobody was willing to seriously discuss this with me and it was the first time that I had to conclude that adults are less smart than I initially thought.

A similar problem was caused by my observation that one cannot make valid statements about non-existent things, e.g. the sentence "pink unicorns do not exist" has no meaning, because by definition there is nothing it could be about.

I was convinced for a while that the two problems are related but I am not so sure about this anymore. (*)

I really do regret that the internet did not exist yet when I was a kid, because it would have helped me to refine my worries tremendously (x) and e.g. think about the 'other worlds strategy' earlier (btw I discussed one problem with the idea of other worlds here). On the other hand, I am glad, for obvious reasons, that it did not exist then.

Of course, writing about a non-existent internet is problematic, so instead let me end this blog post with a link to a rather short proof of the famous equation 1+1=2, a proposition which is occasionally useful.

(*) We leave the question whether I really am (or was) the same person as I was (or am?) as a child for another time(!).

However, then I was convinced that if n is a non-existent thing, e.g. the integer n which solves the equation n*n=7, it is not necessarily true that n=n and therefore the concept of mathematical identities is problematic because of non-existent things. But I cannot be sure that I remember this argument from my childhood correctly.

(x) E.g. with a link to Black's symmetric universe.

added later: If you followed this link, then perhaps you already saw also Rene Descartes' argument about human identity: I cannot doubt that I am, however I can doubt that my body (including my brain) exists; it follows that I am not (just) my body.

This conclusion, approximately 350 years before

*The Matrix*, about the mind-body problem was also the first important contribution to the Copenhagen interpretation imho.### the sound of silence

Now that we all have solved the puzzle from Singapore (or at least read the solution), it is time to think about another one; fortunately it is much easier.

Your boss wants to fire two of three employees, unfortunately you are one of them, and only keep the smartest. This is her challenge: There are 2 black hats and 3 white ones. You three are blindfolded and one hat placed on each of your heads (the two unused hats discarded). After the blindfolds are removed you cannot see your own hat, only the other two. The first employee to figure out the color of his own hat keeps the job.

After the blindfolds are removed you see that the other two guys have white hats. There is an awkward moment of silence and that silence in fact tells you the color of your hat.

What is it?

Your boss wants to fire two of three employees, unfortunately you are one of them, and only keep the smartest. This is her challenge: There are 2 black hats and 3 white ones. You three are blindfolded and one hat placed on each of your heads (the two unused hats discarded). After the blindfolds are removed you cannot see your own hat, only the other two. The first employee to figure out the color of his own hat keeps the job.

After the blindfolds are removed you see that the other two guys have white hats. There is an awkward moment of silence and that silence in fact tells you the color of your hat.

What is it?

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