Discourses about identity
When is one thing equal to some other thing?
I find generalised discourses about identity captivating. When do two entities become one? When can we consider one thing equivalent to another? These are the very questions that gave birth to philosophy.
♾️ When is one thing equal to some other thing?
When is one thing equal to some other things? Barry Mazur's 2007 article offers a compelling perspective on mathematicians' exploration of 'the awkwardness of equality.' It serves as a concise introduction to category theory—a complex and fascinating topic that has, curiously, gained traction in hipster artistic circles. Test yourself and see how far you can delve into it
I found a simple explanation of group theory (Quanta). This is a must read if you want to be my friend.
Understanding the precise language of symmetry is priceless. It’s not just a useful intellectual accomplishment. Personally, it’s one of my proudest (life) achievements. Give it a try!
💻 What’s artificial learning about?
In Computer Science what comes to mind in this day and place is Generalised Unsupervised Learning. Here’s Ilya Sutskever in 2023 explaining some older OpenAI results from 2016.
This is not a state-of-art marketing presentation about LLMs. It’s rather a good (technical) ride to understand why OpenAI succeeded so far. Learn how its chief scientists were thinking about confusing foundational questions, what mental maps they found interesting to navigate the topic.
You’ll encounter topics such as distribution matching, algorithmic mutual information and Kolmogorov complexity. These are precise ways to formalise the notion of “shared structure” across two objects in informatics.
Ilya argues about how to ground theoretically the task of unsupervised learning, which lacked the same rigorous basis of supervised tasks, which can be viewed as convergent approximation problems. His viewpoint shifts the scene: unsupervised learning is a compression problem. This is the fundamental intuition behind LLMs (cf. LLMSs as blurry JPEG of the web). It’s worth to keep it in mind.
Here’s a (meta) example that can help us see what I mean by two “things” having the same structure. Think about prediction and compression. The two tasks share a common structure. Both have to capture patterns to reduce the amount of information needed to describe an (T+1) outcome. The outcome could be the decompressed item or the next entry of a time series. If a whole dynamic process can be compressed and explained in fewer simpler terms, then it can be predicted.
Stories, like equations, are compact explanatory devices. Their compression is their predictive power.
Talking about fundamental dualities…
🔥 The Unreasonable effectiveness of Physics
Read the The Unreasonable effectiveness of Physics (Nautilus). While it is well known that mathematics have allowed tremendous progress in the natural sciences, sometimes it goes the other way around.
The sort of math that emerges from studying reality is the sort our brains tend to like.
A recent paper by Daniele Molinini (UChicago, 2022) explores the converse of the classic question that Eugene Wigner asked in 1960: why is mathematics so effective explaining the natural world?
The sharpest knowledge-seekers out there test their pattern recognition skills continuously. It’s out there in the wild that new analogies, relationships, mappings, dualities emerge. Some of them stick around.
After all, there’s no true mathematics. Just mathematics that mathematicians like.
Once it is possible to translate any particular proof from one theory to another, then the analogy has ceased to be productive for this purpose; it would cease to be at all productive if at one point we had a meaningful and natural way of deriving both theories from a single one. … Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections, their furtive caresses, their inexplicable quarrels; alas, all is just one theory, whose majestic beauty can no longer excite us. Nothing is more fecund than these slightly adulterous relationships; nothing gives greater pleasure to the connoisseur, whether he participates in it, or even if he is an historian contemplating it retrospectively, accompanied, nevertheless, by a touch of melancholy. The pleasure comes from the illusion and the far from clear meaning; once the illusion is dissipated, and knowledge obtained, one becomes indifferent at the same time. … (A. Weil, De la métaphysique aux mathématiques, Science 60, p. 52–56, emphasis mine).
This passage from a letter of André Weil to his sister Simone is what made me fall in love with maths.
Hope you enjoy it,
David
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