For a while, I was confused about the differences between subjects like manifold calculus, differential geometry, and differential topology. This post discusses my current understanding of how these subjects are defined relative to each other.
Tag: Studies
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The Transcendentality of Normal Numbers
Here, we investigate a conjecture I have, that all normal numbers are transcendental.
In this post, we summarize my previous work of the series in a more readable and concise manner. (Proofs are not included.)
In this post, we continue the discussion from the previous in the series.
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An Abstract Approach to Generating Sets 3
In this post, we continue the discussion from the previous in the series.
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The Falsifiability of Newton’s Laws
Consider the following thought experiment: we setup an experiment consisting initially of objects and known forces, and we run Newton’s laws to predict the motion of the objects. We then run the experiment, and the observed results do not match the predicted calculations. Can we just attribute the discrepancy to a new force that we haven’t yet studied, and still satisfy Newton’s laws with this just-defined unknown force? Can we explain all discrepancies with Newton’s laws in this way? Then, are the laws really saying anything materially? Can we think of forces as convenient mathematical devices to represent “contributions of mass times acceleration”?
In other words, taken away from the specific laws of gravity and the other forces, are Newton’s laws themselves falsifiable?
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A Curious Question on Modular Arithmetic
In this post, we investigate a curious question: say we are given 
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An Abstract Approach to Generating Sets 2
We continue the discussion from this post.
We continue the discussion from the previous post in the series.
Topologies and measure theory’s sigma-algebras look superficially similar, but the differences in their defining axioms lead to differences in their resulting study. In this post, we write this out to understand this better.
