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Examples in Mathematics

I don't think it is strictly true that Grothendieck didn't rely on examples. Here is a quote from a letter from Luc Illusie, who was a grad student under Grothendieck. Quote:

“In his filing cabinets, located behind his desk, Grothendieck kept many handwritten notes, where he had studied specific examples: he sometimes told me that he was weak on surfaces, but as everybody knows, he was not so weak in local algebra, and he knew enough of curves, abelian varieties and algebraic groups to be able to test his ideas. Also, his familiarity (and constant interest) in analysis and topology was a strong asset. All these examples appeared when you discussed with him."

Slightly related ... there is a recent philosophy of mathematics book (albeit a work of continental philosophy, but it is still interesting). I say slightly related, because firstly it has an entire chapter on Grothendieck and creativity, and secondly the entire book is about the interplay between examples and theory. It goes into the way mathematics moves up and down a ladder of abstraction (from particulars to universals, local to global, specialization to generalization). It focuses on the last 100 years of mathematics, concentrating on algebraic geometry, category theory, model theory, and others. The author splits up the abstractions into:

  • Eidal mathematics, or "transfusions of form." This is mathematics that moves up the ladder from the particular to the universal. In this part of the book it focuses on the work of Serres, Langlands, Lawvere, Shelah.
  • Quiddital mathematics, or "transfusions of reality." This is abstractions made concrete moving down the ladder. Here the book focuses on Atiyah, Lax, Connes, Kontsevich.
  • Archeal mathematics, or "decantations of the universal." This is mathematics that studies the invariants while one moves up and down the ladder. Here it focuses on Freyd, Simpson, Zilber, Gromov.

Now you could say this philosopher is just dressing up stuff already known to mathematicians (Both Kevin Houston's How to think like a mathematician, and Mason/Burton's Thinking Mathematically talk about generalization and specialization). But it is still interesting, as I think it may be the only philosophy of mathematics book out there that does an indepth treatment of 20th and 21st Century mathematics that goes beyond Russell and Frege.

Open Thread, December 2-8, 2013

Semi related article (pdf link):

What Is the Enemy of My Enemy? Causes and Consequences of Imbalanced International Relations, 1816–2001


This study explores logical and empirical implications of friendship and enmity in world politics by linking indirect international relations (e.g., “the enemy of my enemy,”“the enemy of my friend”) to direct relations (“my friend,”“my enemy”). The realist paradigm suggests that states ally against common enemies and thus states sharing common enemies should not fight each other. Nor are states expected to ally with enemies of their allies or with allies of their enemies. Employing social network methodology to measure direct and indirect relations, we find that international interactions over the last 186 years exhibit significant relational imbalances: states that share the same enemies and allies are disproportionately likely to be both allies and enemies at the same time. Our explanation of the causes and consequences of relational imbalances for international conflict/cooperation combines ideas from the realist and the liberal paradigms. “Realist” factors such as the presence of strategic rivalry, opportunism and exploitative tendencies, capability parity, and contiguity increase the likelihood of relational imbalances. On the other hand, factors consistent with the liberal paradigm (e.g., joint democracy, economic interdependence, shared IGO membership) tend to reduce relational imbalances. Finally, we find that the likelihood of conflict increases with the presence of relational imbalances. We explore the theoretical and practical implications of these issues.

November 2013 Media Thread

The Logic Matters blog has three posts on the topic "Does mathematics need a philosophy?"

Part One, Part Two, Part Three.

November 2013 Media Thread

The page What Has Experimental Philosophy Discovered about Expert Intuitions? on the Experimental Philosophy blog has been updated. It contains several links to papers on the area.

How to Become a 1000 Year Old Vampire

You may remember early in the Arabian revolutions in Libya, an American student took the summer off college to fight in the revolution.

Adding to this, there is an entire online community of these people at the Black Flag Cafe. Outsiders label them as "war tourists," but the majority of them are journalists, war photographers, businessmen, and humanitarians/activists. The website was founded by Robert Young Pelton (whose wikipedia page is worth reading). He wrote a great book, that is filled with practical information.

One way to manipulate your level of abstraction related to a task

RE: how / why distinction.

John Whitmore's book Coaching for Performance goes into this. Snippit from the book:

If commanding a person to do what they need to do does not produce the desired effect, what does? Let's try a question.

  • "Are you watching the ball?" How would we respond to that? Defensively, perhaps, and we would probably lie, just as we did at school when the teacher asked us if we were paying attention.

  • "Why aren't you watching the ball?" More defensiveness -- or perhaps a little analysis if you are that way inclined. "I am," "I don't know," "because I was thinking about my grip," or, more truthfully, "because you are distracting me and making me nervous."

These are not very effective questions, but consider the effect of the following:

  • "Which way is the ball spinning as it comes toward you?"
  • "How high is it this time as it crosses the net?"
  • "Does it spin faster or slower after it bounces, this time, each time?"
  • "How far is it from your opponent when you first see which way it is spinning?"

These questions are of an altogether different order. They create four important effects that neither the other questions nor commands do:

  • This type of question compels the player to watch the ball. It is not possible to answer the question unless he or she does that.

  • The player will have to focus to a higher order than normal to give the accurate answer the question demands, providing a higher quality of input.

  • The answers sought are descriptive not judgmental, so there is no risk of descent into self-criticism or damage to self-esteem.

  • We have the benefit of a feedback loop for the coach, who is able to verify the accuracy of the player's answers and therefore the quality of concentration.

July 2013 Media Thread

SEP has a new article up on Analogy and Analogical Reasoning.

July 2013 Media Thread

Tychomancy: Inferring Probability from Causal Structure by Michael Strevens (a philosophy professor at NYU). From the blurb:

Maxwell's deduction of the probability distribution over the velocity of gas molecules—"one of the most important passages in physics" (Truesdell)—presents a riddle: a physical discovery of the first importance was made in a single inferential leap without any apparent recourse to empirical evidence.

Tychomancy proposes that Maxwell's derivation was not made a priori; rather, he inferred his distribution from non-probabilistic facts about the dynamics of intermolecular collisions. Further, the inference is of the same sort as everyday reasoning about the physical probabilities attached to such canonical chance setups as tossed coins or rolled dice. The structure of this reasoning is investigated and some simple rules for inferring physical probabilities from symmetries and other causally relevant properties of physical systems are proposed.

Not only physics but evolutionary biology and population ecology, the science of measurement error, and climate modeling have benefited enormously from the same kind of reasoning, the book goes on to argue. Inferences from dynamics to probability are so "obvious" to us, however, that their methodological importance has been largely overlooked.

There is also a brief chapter summary here.

July 2013 Media Thread

Logic: The Drill by Nicholas J.J. Smith and John Cusbert.

Free 300 page PDF that contains a variety of solved exercises from propositional and predicate logic. Description from the authors:

One obvious use of this work is as a solutions manual for readers of Logic: The Laws of Truth—but it should also be of use to readers of other logic books. Students of logic need a large number of worked examples and exercise problems with solutions: the more the better. This volume should help to meet that need.

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