In this post, we continue the discussion from the previous in the series.
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Author: Nihal Uppugunduri
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An Abstract Approach to Generating Sets 4
In this post, we continue our discussion from the previous in the series.
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Constructing Compatible Operations
Recently in my ring theory class I heard about the theorem that if a ring satisfies 
Recently, I have been thinking about the unreasonable effectiveness of definitions in mathematics. Without making any additional assumptions or knowing any additional information about a situation, by simply making judicious definitions and studying their theory, we can get a lot farther than we could without those definitions. (As you might have noticed, “unreasonable” here is a nod to Eugene Wigner’s classic piece.) Upon further thought, it seems somewhat baffling that this works so well, especially in certain cases.
One of the things I learned later on in my physics classes was that physical laws don’t necessarily guarantee uniqueness. Of course, when written out, this sounds obvious — why would we expect a particular set of laws to always guarantee only one satisfying physical state? — but intuitively I didn’t recognize this until later on.
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Avoid Oversimplified and Faulty Logic
Many concepts in subjects like math and science can be very technical, requiring lots of background knowledge to understand. This limits the audience of people who can discuss and work with these concepts. But given how much these subjects can influence our society, for example with the modern tech industry and computer science, there is a real need for more people to understand these concepts. Simplified explanations can go a long way towards enabling this. However, it can be hard to write such explanations while also describing the concepts accurately and incorporating scientists’ and engineers’ input on how to present their own work.
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The Physical Existence of Math
Can math be ascribed some sort of a physical existence? How could this work, from a philosophical perspective?
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Separating Mathematical Logic and Set Theory
Many authors consider “mathematical logic and set theory” to be one subject, dealing with the study of concepts that are deemed fundamental to math. However, I disagree with this, for a number of reasons I will outline now.
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Is Asymptotic Analysis Still Valuable?
Many software engineers who have been in the industry seem to view asymptotic analysis (“big-O”) as not as effective or relevant to their work. Yet it is still fundamentally emphasized in academic computer science programs, forming a major part of standard algorithms and data structures courses. Is there still some value in asymptotic analysis, or is it better to evolve away from it going forward?
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Reducing Dependence on Credentials in Math
Ostensibly, at the end of the day, math should be just about what can be proven right from wrong. Thus, we may reason that we shouldn’t care about someone’s credentials in order to cite them — we can just read their proof and check the validity of their words ourselves.
But despite this, I have often found myself evaluating the authoritativeness of authors through things like credentials when trying to pick resources to learn from. Why do I end up doing this? Are there in fact aspects of doing math for which credentials become important? For these aspects, can we devise new methods to reduce dependence on credentials, or even eliminate it, in favor of judging just based on content?