Mathematics Without Numbers (1959)

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Aren’t many algebraic results dependent on counting/divisibility/primality etc...?

Numbers are such a fundamental structure. I disagree with the premise that you can do mathematics without numbers. You can do some basic formal derivations, but you can’t go very far. You can’t even do purely geometric arguments without the concept of addition.

Kemeny is an interesting fellow. He is part of the duo responsible for the BASIC language (at Dartmouth).

I found his book "Man and the Computer" particularly prescient.

https://en.wikipedia.org/wiki/John_G._Kemeny

https://archive.org/details/mancomputer00keme

Addition does not require numbers. It turns out, no math requires numbers. Even the math we normally use numbers for.

For instance, here is associativity defined on addition over non-numbers a and b:

a + b = b + a

What if you add a twice?

a + a + b

To do that without numbers, you just leave it there. Given associativity, you probably want to normalize (or standardize) expressions so that equal expressions end up looking identical. For instance, moving references of the same elements together, ordering different elements in a standard way (a before b):

i.e. a + b + a => a + a + b

Here I use => to mean "equal, and preferred/simplified/normalized".

Now we can easily see that (a + b + a => a + a + b) is equal to (b + a + a => a + a + b).

You can go on, and prove anything about non-numbers without numbers, even if you normally would use numbers to simplify the relations and proofs.

Numbers are just a shortcut for dealing with repetitions, by taking into account the commonality of say a + a + a, and b + b + b. But if you do non-number math with those expressions, they still work. Less efficiently than if you can unify triples with a number 3, i.e. 3a and 3b, but by definition those expressions are respectively equal (a + a + a = 3, etc.) and so still work. The answer will be the same, just more verbose.

That is not really a very deep result.

>Numbers are just a shortcut for dealing with repetitions

An interesting explanation, I think I agree

Please don't do this here. Article summaries have always been eschewed on HN.

I would have liked a summary before reading.

Why is writing a summary a bad thing?

Because HN readers can't know if the summary is an accurate representation of the original article, nor what detail or nuance has been winnowed out in the summarizing process. But if there is a summary that seems "good enough" to form an opinion, then the discussion on HN will be based on the summary, not on the complete article. We see the same thing with editorialized titles.

A better way to get a taste of the article is to look over the HN discussion. The top comment(s) should give people a hint as to what it's about and whether it's worth the time to read the whole thing. Otherwise just reading the HN discussion should be a good way to get the jist of it. But that only works if enough of the commenters have actually read the whole article rather than a summary.