> The theory of finite fields is based on the theory of prime numbers, because the finite fields are sets of residues modulo a prime number or modulo a power of a prime number.
It is note quite correct that the finite field of order p^k is the set of residues modulo p^k when k > 1. Instead this field is obtained as a splitting field of the field of order p (which is the set of residues mod p).
Agreed, thanks for the clarifications. Another result worth mentioning, which also shows that you cannot hope to uniquely characterize a structure by "reasonable" axioms is the Löwenheim–Skolem theorem which predates Godel's incompleteness (although the history of these results is somewhat convoluted).
There, the obstacle is in some sense of a simplest nature: if your set of axioms admits a countable model, then it admits models of all infinite cardinalities. In other words, it shows that there is something fundamentally impossible in trying to capture an infinite structure (like numbers) by finite means (e.g. recursively axiomatizable).
This idea of giving "meaning" to a set of axioms is precisely captured by the notion of "interpretation" in logic [1]. The rough idea is to map the symbols of the formal language to some pre-existing objects. As you say, this gives one way of formalizing truth: a sentence (string of symbols that respect the syntax of your language) is true if it holds for the objects the sentence is referring to (via a chosen interpretation). This notion of truth is sometimes referred to as semantic truth.
An alternative approach is purely syntactic and sees a logical system as collection of valid transformation rules that can be applied to the axioms. In this view, a sentence is true if it can be obtained from the axioms by applying a sequence of valid transformation rules. This purely syntactic notion of truth is known as “provability”.
Then the key question is to ask whether the two notions coincide: one way to state Godel's first incompleteness theorem is that it shows the two notions do not coincide.
The article is a bit oversimplifying in summarizing the axiomatic crisis as being problem with sentences like “this statement is false“.
This being said, your intuition is absolutely correct, the crux of the issue is with ‘this‘. What mathematicians realized is that if you are not careful with your choice of axioms, the resulting logical system becomes too “powerful” in the sense that it becomes self-referential: you can construct sentences that refer to themselves in a self-defeating manner.
As others have mentioned, this is the idea underlying Gödel's incompleteness theorem but also, to some extent, Russel's paradox that came before and is what the article is referring to. In Russel's paradox, the contradiction comes from constructing the set of all sets that contain themselves.
Isn't this what is suggested below the second picture?
> Really, it’s harder than this picture suggests, because many experiences are based on other students. If I want you as my project partner but you want to forget I exist, then something has to give.
I agree. The GP comment contains some inaccuracies: most of the spaces of functions considered in functional analysis do not have an inner product defined on them, but are still vector spaces. The existence of an inner product presupposes a vector space structure, but the converse is not true…
Perhaps the most famous example is provided by the Lp spaces [1] consisting of functions whose pth power is absolutely integrable. For p≥1, these spaces are Banach spaces (complete normed spaces) but it is only when p=2 that the norm is associated with an inner product.
The realization that functions can be treated as elements in an abstract vector space (with infinitely many dimensions) is a turning point in the history of mathematics that led to the emergence of the sub-field known as functional analysis.
The significance of this paradigm shift is that it allowed mathematicians to apply some of the geometric intuition developed from the study of finite-dimensional spaces (such as the 3D Euclidean space) to difficult questions involving functions, such as the existence of solutions to certain differential equations.
The history of this change of perspective is absolutely fascinating and can be traced back to the end of the 19th century and beginning of the 20th century. At the time, work on axiomatic foundations of mathematics was driving a systematization of the study of mathematical objects by capturing their structure with a concise list of axioms. This is for example how the concept of an abstract vector space was born, encompassing not only Euclidean spaces but also infinite-dimensional spaces of functions.
An early reference already demonstrating this change of perspective, albeit in a primitive form, is a memoir by Vito Volterra from 1889 [1]. The PhD thesis of Maurice Fréchet from 1906 [2] is arguably the work that was most influential in crystalizing the new paradigm and presenting it in a modern form that served as a key reference for the first half of the 19th century. Of course, these are only two among a multitude of works around that time. Looking at later developments in the 19th century, it is hard not to also mention the book by Stefan Banach from 1932 [3].
Among many other things, SUB Göttingen maintains the Göttingen Center for Retrospective Digitization (GDZ) [1] which hosts a very impressive collection of digitized documents. It seems that the managerial changes that prompted the open letter could pose a serious threat to the project.
Regarding stochastic gradient descent, I think there has been an increased understanding in recent years, that the randomness introduced by the random sampling/batching is not only helpful in reducing the computational cost (compared to computing the full gradient) but also in adding noise to escape local minima. Some variants of stochastic gradient descent in fact add some additional random noise to amplify this latter effect and some theoretical guarantees have started to emerge.
It is note quite correct that the finite field of order p^k is the set of residues modulo p^k when k > 1. Instead this field is obtained as a splitting field of the field of order p (which is the set of residues mod p).