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Wes Hansen
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The question posed was, "When can mathematics be wrong?" My answer:

To begin with, since the advent of non-Euclidean geometry, mathematics has been divided into two categories: pure and applied. In both cases, when mathematics is wrong, erroneous proof in the case of pure, erroneous model in the case of applied, it is invariably the result of human error. These errors can take a number of different forms, too many to list. On the pure side, this is the motivation for peer review and automated proof checking; on the applied side, this is the motivation for peer review and empirical verification.

As an example on the pure side, human error recently became obvious to me due to my own work in Model Theory. I invented a counter-example to the definition of a recursive structure! This is quite obviously a profound human error caused, or so I believe, by expectation bias.

The definition I use in my counter-example paper comes from a paper by Samuel Reid who copies it from Richard Kaye’s Models of Peano Arithmetic. It is Definition 9 in the Reid paper:

Definition 9. An L_A-structure M is recursive if and only if there are recursive functions ⊕ : N^2 → N, ⊗ : N^2 → N, a binary recursive relation <⊆ N^2 and natural numbers n_0, n_1 ∈ N such that {N, ⊕, ⊗, <, n_0, n_1} ∼= M.

But basically what this says is that a structure is recursive iff it is ring isomorphic to the “standard” model of PA! In short, Definition 9 is not a definition, rather, it is a conjecture - a conjecture which I prove false by construction! In my counter-example paper I establish set theoretic foundations for my structure in ZFC/AFA but you can think of the structure as {N x N, <, +, *, (1, 0), (0, 0)} with lexicographic order, coordinate-wise addition, and multiplication defined by:

(a, b) * (c, d) = (a * c, b * c + a * d + b * d).

It is my belief that this structure was not invented sooner primarily because of Gauss’ ordered pair representation of the complex field; this was, of course, motivated by a silly belief in Platonism or mathematical realism - Gauss was motivated by a desire to “remove metaphysical difficulties from mathematics.”
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It seems to me that the Last Question is always, always, "Would you do it again?," or its equivalent, "Was it worth it?"
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If you haven't read "The Development of Mathematics" by E. T. Bell yet, Rapaport, you might really enjoy it! It was first published in 1940, reissued in 1945, but Amazon still lists it for $20 on Kindle! That should indicate how good it is . . .
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I like tea . . .
Tea and mindfulness are a natural pairing: the ritual of tea can be a convenient focal point to draw your attention to the present moment, and by being more present, you can appreciate the tea on a deeper level.

We hope this article will also draw attention to some of the conscious design choices behind the RateTea website, including why we classify teas by style and region (in large part to help people become more aware of where their tea comes from) and why we avoid "busy" or "addictive" features like the notifications and animated feeds that are unfortunately the norm for social media these days. #tea #mindfulness
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I follow a Brazilian entomologist, Eduardo Fox, on Quora and he writes some really interesting answers! I recently commented on one of his answers and he linked me to this cockroach egg-laying video; here's the short comment thread:

Me: Yeah, wow, have you ever watched a roach egg hatch? The roach eggs are so small but the baby roaches, they just keep coming and coming . . .

Dr. Fox: I have worked with cockroaches for several years, on biological control. Yes, I did see them hatch quite a few times. I am linking a video at the end of this comment. Actually we don’t get to see the roach eggs, but their egg cases (oothecae) which are fairly large, like beans. The 1st time I witnessed them hatching I was flabbergasted as they push out from their case like a glove, jointly. Didn’t look like roaches while doing it. From 3 mins, in the video below.
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Here's a recipe, but there's better . . .
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What's the best tea for making Tibetan butter tea?
There are compelling advantages to buying, brewing, and drinking loose-leaf tea, including access to the best teas, better control over brewing, much lower price for teas of similar quality, and promoting sustainability through reducing packaging. #tea #sustainability #loosetea
Loose Tea
Loose Tea
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