Heuristics Lead to Rigour

As you learn more ideas in mathematics, it’s easy to start feeling like certain ideas are “below” you. This often comes in the form of saying that ideas are “trivial”, as if they shouldn’t take up any of your time. This can be exacerbated further in mathematics by the idea of rigour. Once we learn that not all proofs are equal, it can be tempting to say, “Okay, I get this proof, but that’s not…

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Colour, psychophysics, and the scientific vs. manifest image of reality

Recently on TheEGG, I’ve been writing a lot about the differences between effective (or phenomenological) and reductive theories. Usually, I’ve confined this writing to evolutionary biology; especially the tension between effective and reductive theories in the biology of microscopic systems. For why this matters to evolutionary game theory, see Kaznatcheev (2017, 2018). But I don’t think that microscopic systems are the funnest place to see this interplay. The funnest place to see this is in…

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Knowledge From Repetition

When I’m sitting in class and listening to the professor, I get frustrated when they go over a concept once and then continue on as if everything is clear. I think to myself, “Do they really expect us to understand a concept from just one example?” And yet, I realize that I do the same thing when I’m working with a student. This is a bit disconcerting, and it reminds me that teaching is a…

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I Could Be Wrong

As a physics student, I’m taught over and over again that science is about checking to see where we have made incorrect assumptions. The goal of science is to correct our assumptions about the world, and find better descriptions for what is going on. Of course, this is a distilled version of the goal of science (and I imagine some would disagree), but my point is that science aims to perform a consistency check on…

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Constant-sum games as a way from non-cell autonomous processes to constant tumour growth rate

A lot of thinking in cancer biology seems to be focused on cell-autonomous processes. This is the (overly) reductive view that key properties of cells, such as fitness, are intrinsic to the cells themselves and not a function of their interaction with other cells in the tumour. As far as starting points go, this is reasonable. But in many cases, we can start to go beyond this cell-autonomous starting point and consider non-cell-autonomous processes. This…

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Where Does It Belong In The Web?

If you look at a typical mathematics class, you will find that they follow a similar rhythm. First, students are presented with a definition. A few examples of how that definition applies might be given, and then the rest of the class is spent proving results based on this definition. It’s a familiar recipe, and it works well for transmitting information. If you asked students to give a presentation on the topic, I would predict…

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Being Good At Mathematics

“How did you get so good?” This is a question I’m asked from time to time with respect to mathematics and physics. People see the kinds of grades I get, and they want to know what my secret is. I presume they think I have a method, or at least some explanation as to why I get great grades at school. When I hear this question, I often have to hold myself back from ranting…

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Local maxima and the fallacy of jumping to fixed-points

An economist and a computer scientist are walking through the University of Chicago campus discussing the efficient markets hypothesis. The computer scientist spots something on the pavement and exclaims: “look at that $20 on the ground — seems we’ll be getting a free lunch today!” The economist turns to her without looking down and replies: “Don’t be silly, that’s impossible. If there was a $20 bill there then it would have been picked up already.”…

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Clarity and Words

A cornerstone of physics and mathematics is the process of labeling. If you want to learn any subject, you need to get familiar with the jargon of the field. In physics, terms like “work”, “resistance”, “capacitance”, “energy”, “potential”, and many more have precise meanings. Likewise, the terms “function”, “continuous”, and “one-to-one” have precise meanings in mathematics. When you first start learning the subject, it’s easy to be overwhelmed by the sheer number of terms. By…

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