Last week, a student wrote me, “Is there going to be a proof question on the number theory exam?”
I answered,
As far as I’m concerned, the answer to every mathematical question is a proof, because everybody can check whether the answer is right.
I meant that the answer should provide the means for the reader to re-enact the answerer’s thought.
View from Büyükdere, Sarıyer, Istanbul
We live near the big building at Hacıosman
just over the horizon on the right
Sunday, June 30, 2024
This is about an image that is intended
In the Platonic dialogues, Socrates frequently mentions τέχνη (technê), which is art in the archaic sense: skill or craft. The concern of this post is how one develops a skill, and what it means to have one in the first place.
After Wordle appeared, a number of variants came out. One of the least popular may be Zpordle, or ℤp-ordle. I imagine it could be more popular, if people knew it did not require advanced mathematics. It still involves numbers, to which some people declare an allergy. Nonetheless, I think Zpordle can be explained in elementary-school terms, and that is what I shall try to do here.
Screenshot of the Zpordle game won on May 27, 2022
This attempt at exposition of Gödel’s Incompleteness Theorem was inspired or provoked by somebody else’s attempt at the same thing, in a blog post that a friend directed me to. I wanted in response to set the theorem in the context of mathematics rather than computer science.
This post is a response by Alexandre Borovik to my previous post. The following words then are Sasha’s:
Dear David,
I joined the AMR. In my view, its areas of activities are sufficiently clearly defined [on its homepage]:
The AMR has several initiatives under development. These include:
- AMR colloquia, lectures and workshops, exploring new ways to present research
- Updates and reviews of new research
- Reviews of classic influential papers
- Discussions of open problems
- Video expositions of mathematical research
- AMR journals and publications leveraging new technological opportunities
- Interviews of mathematicians
- Developing new ideas for the flourishing of the international mathematical research community
This post concerns the Association for Mathematical Research, or AMR. A number of people are upset by its existence. I am not exactly one of them, but am suspicious, mainly because I do not know why a new organization would be needed, when we already have
The Twitter account of the AMR is dated to April, 2021. The website of the AMR supplies a list of founding members, but no account of when, how, or why they became founders. The site has a brief mission statement:
THE MISSION of the AMR is to SUPPORT MATHEMATICAL RESEARCH and SCHOLARSHIP
Are those other organizations not doing a good job? Continue reading
To this post, I am adding this introduction in July 2021. I have returned to some of the ideas of the post, and I see that I left them in a jumble. They may still be that, but I am trying to straighten up a bit.
Beyond this introduction, the post has three parts. Part III takes up more than half of the whole post and consists of my notes on
In Anderson’s article I see – as I note below – ideas that are familiar, thanks to my previous reading of philosophers such as Robin George Collingwood, Mary Midgley, and Robert Pirsig. Henry David Thoreau may not exactly be one of those philosophers, but he is somehow why I came to write this post in the first place.
Here is a table of contents for the whole post:
Note added May 19, 2026. I return to this post now, because the question arises with a friend of whether the US should be at war with Iran. This post does not actually answer that or any such question. It would, implicitly, if it said that war was never the answer, or war never solved any problem. This would absolve us from needing to think about any particular case, such as whether the American colonists should have taken up arms against the British Crown, or John Brown should have done the same against slavery. This post is mainly a response, by a mathematician and a product of a great-books program, to contentious or sententious things said by fellow academics. Concerning international relations as such, see the post referred to below, “External Politics.”
Pacifism is properly pacificism, the making of peace: not a belief or an attitude, but a practice. Mathematics then is pacifist, because learning it means learning that you cannot fight your way to the truth. Might does not make right. If others are going to agree with you, they will have to do it freely. Moreover, you cannot rest until they do agree with you, if you’ve got a piece of mathematics that you think is right; for you could be wrong, if others don’t agree.
Large parts of this post are taken up with two subjects:
The notion (due to Collingwood) of criteriological sciences, logic being one of them.
Gödel’s theorems of completeness and incompleteness, as examples of results in the science of logic.
Like the most recent in the current spate of mathematics posts, the present one has arisen from material originally drafted for the first post in this series.
In that post, I defined mathematics as the science whose findings are proved by deduction. This definition does not say what mathematics is about. We can say however what logic is about: it is about mathematics quâ deduction, and more generally about reasoning as such. This makes logic a criteriological science, because logic seeks, examines, clarifies and limits the criteria whereby we can make deductions. As examples of this activity, Gödel’s theorems are, in a crude sense to be refined below, that
everything true in all possible mathematical worlds can be deduced;
some things true in the world of numbers can never be deduced;
the latter theorem is one of those things.
