How small is an atom? Well, a scientist would probably answer that the radius of a typical atom is one tenth of a bilionth of a meter, and that the biggest atom (cesium) is approximately nine times the smallest atom (helium). As far as I know (not much, to be honest…), the answer is right; but it will hardly satisfy the curious child inside each of us. Continue reading
The new-born FilMat italian network for the philosophy of mathematics announces its first international conference on the topic “Philosophy of Mathematics: objectivity, cognition, and proof”. The conference will take place at San Raffaele University, Milan, on 29-31 May 2014.
The FilMat — of which I am proud to be a member — was created few months ago by a group of Italian scholars in philosophy of mathematics , originally met at the Scuola Normale Superiore in Pisa at the 2012 conference “Philosophy of Mathematics: from Logic to Practice”, and aims to foster the gathering of scholars working either in Italy or abroad on the philosophy of mathematics and strictly related fields, with special attention to those at early stages of their careers.
One of the milestones of contemporary philosophy of mathematics is Benacerraf‘s 1965 article “What Numbers Could Not Be“.1 There he offers a compelling argument according to which numbers cannot be considered as sets — namely, they cannot be metaphysically identified with sets.
The argument is quite simple: if numbers were sets, we should be able to find a unique progression of sets with which numbers can be identified. But this is apparently impossible: there is a lot of ω-series that can serve as well for the aim. For example, we can adopt von Neumann’s series, and say that , and so on, where the successor function is defined by . Or we can adopt Zermelo’s series, and say that , and so on, where the successor function is defined by . Now, the problem is: is or is ? Benacerraf presents then the example of two children, Ernie and John. The first learned that von Neumann’s ordinals are the natural numbers, while the latter that Zermelo’s ordinals are the natural numbers. Now, they will be easily able to learn arithmetic set theoretically via the above constructions, and they will agree on any arithmetical theorem, except that for Ernie it is true that , while for John it is false!
[The following post is based on a seminar given at the Scuola Normale Superiore (Pisa) in June 2011. Here you can find the complete text of the talk (in Italian).]
In 1946, Paul Artur Schilpp asked Kurt Gödel to write a paper to be included in the collective volume on Albert Einstein for the “Library of Living Philosophers” collection (edited by Schilpp himself). As shown by a letter from Schilpp to Gödel dated July 10th 1946, Gödel himself should have had to propose an argument for the paper, but he never replied. Schilpp, then, suggested he write an article on the topic: “The realistic standpoint in physics and mathematics”. But Gödel was hesitant: he thought himself to be insufficiently expert on the topic to worthily contribute. At the most, he considered himself able to contribute some considerations on the notion of time as resulting from Einstein’s theory of relativity and on the relationship between this and the idealistic thesis of the non-existence of objective time. Schilpp was enthusiastic and thus, three years later (in 1949), a short article was published under the title “A remark about the relationship between relativity theory and idealistic philosophy”1
Are you enjoying your Easter egg? Well, have you ever tried to draw an egg? Have you ever wondered about its shape? It is not a circle, it is not an oval — what is that? It’s an egg-curve! Ok, but how can we draw it and how can we define it in mathematical terms? Here you can find a lot of different ways to do that!
But maybe you should just enjoy your chocolate eggs and don’t think too much…
Next week I will be in Munich for a conference on theoretical terms at the Munich Center for Mathematical Philosophy (MCMP). I will talk about the role of mathematics in a very interesting case of existential prediction in physics: the discovery of the omega minus particle. Here is the website of the conference.
Some time ago, I chanced upon the following puzzle:
Five men find themselves shipwrecked on an island, with nothing edible in sight but coconuts, plenty of these, and a monkey. They agree to split the coconuts into five equal integer lots, any remainder going to the monkey.
Man 1 suddenly feels hungry in the middle of the night, and decides to take his share of coconuts at that very moment. He finds the remainder to be one after division by five, so he gives this remaining coconut to the monkey and takes his fifth of the rest, lumping the coconuts that remain back together. A while later, also Man 2 wakes up hungry, and does exactly the same thing: takes a fifth of the coconuts, gives the monkey the remainder, which is again one, and leaves the rest behind. So do men 3, 4, and 5. In the morning they all get up, and no one mentions anything about his coconut-affair on the previous night. So they share out the remaining lot in five equal parts finding, once again, a remainder of one left for the monkey. Find the initial number of coconuts.1
I won’t give you the answer, so you can enjoy finding it by yourself. The solution is not trivial, for you have to solve a Diophantine equation, but in the end it is not too much complicated. Obviously, there is an infinite number of possible solutions, but the problem implicitly asks for the smallest one.
The puzzle, per se, is not very intriguing, but what makes it interesting is a story that usually goes with it. Continue reading