What should philosophers teach in quantitative reasoning courses?

Most philosophy departments offer some logic course that satisfies a mathematical, quantitative, or formal reasoning general education requirement. My university describes their quantitative and formal reasoning requirement as follows:

To help you develop important analytical skills, these courses focus on the presentation and evaluation of evidence and argument, the understanding of the use and misuse of data, and the organization of information in quantitative or other formal symbolic systems.

Given that these are quantitative and general education requirements, what should we, as philosophers, teach in these courses?

Almost all of these courses choose topics from the following categories:

  • categorical logic (square of opposition, categorical inferences)
  • categorical logic (diagrams)
  • informal fallacies and reasoning
  • propositional logic (proofs)
  • propositional logic (truth-tables)
  • propositional logic (tableaux)
  • inductive and causal reasoning, including causation vs correlation vs explanation
  • probabilistic reasoning, including the probability calculus

Most of them, in my experience, do not proceed to quantification theory. Perhaps the thought is that this much content is too difficult for this level (but categorical logic is an easier, albeit more cumbersome, way to teach universal and existential reasoning).

So: given the quantitative reasoning aims of “organization of information in quantitative or other formal symbolic systems”,  which of these should we select?

I think that categorical logic should be struck from this list. Categorical reasoning is too unwieldy to be justifiable, even if the diagrams are fun. Formulas need not have the categorical form, and they should not be forced into it, even for pedagogical purposes.

Propositional logic, on the other hand, is woefully inexpressive. But it is a useful tool for teaching transferable analytical skills using syntactic and semantic methods. I cannot sing the virtues of proofs, tables, and tableaux too highly for imparting the analytical habit of mind.

What about informal fallacies? I say that they should be struck. They do not, first, teach any analytical skills. They do help one identify mistakes in reasoning, which may then be leveraged to evaluate arguments and assess the use (or misuse) of data. But they do not in themselves demand much development of analytical skills. They are just labels that bring out faulty reasoning more clearly.

Also, they can be introduced and taught in a week. Familiarity with them is the only real virtue of teaching them. Informal fallacies do not seem to be worth assessing using multiple choice tests or projects outside of class. That seems to be more about testing our ability to remember the names of fallacies—which is not a useful thing to teach—as opposed to sharpening our wits so that we can catch mistakes in reasoning.

If we are concerned to teach one to reason well—to cultivate the analytical habit of mind—then it seems to me that we are better off teaching inductive and causal reasoning, plus the probability calculus. These do cultivate active analytical thinking in a way that passively identifying fallacies does not.

So I will be minimizing or leaving out informal fallacies altogether in my critical thinking class! I am quite excited about what the results will be.

What 30 Terms Should Every Intro Phil Student Know?

In thinking about my syllabus for introduction to philosophy, I was struggling with the fact that most freshman college students have not encountered philosophy before. Many professional philosophers are fighting this trend, like the good folks at PLATO and the Iowa Lyceum (and Utah Lcyeum, Western Michigan Lyceum, and SoCal Philosophy Academy).

I came up with the idea of giving them a list of roughly 30 philosophical terms that every introduction to philosophy student should know. I also am giving them an exam on those terms around week three. I am curious: what are your roughly 30 terms that every first-time philosophy student should leave your course knowing?

I won’t share my list because, first, my list is geared slightly towards what we will read in the course, and second, my list might bias the answers. Some terms are fairly easy: they should know what an argument is, for example, and they should probably know what ethics, metaphysics, and epistemology are. That leaves a lot open.

So what would go on your list?

Opening Remarks at Thesis Defense

What are the discipline-wide norms are for thesis defense remarks in philosophy. The following seem like good ways to frame a thesis at the defense: outlining the trajectory of the main argument, bracketing off questions you did not aim to address, and letting your motivation for undertaking the work inject itself a bit. Are there other aims that one might address? At any rate I recently had my thesis defense. I am still excited about the conversation with my committee, so I thought to post my opening remarks here. Enjoy!


Professors, my chief purpose in this work was to answer the Socratic ‘What is it?’ question where the ‘it’ is logical atomism. That involved going back to its roots and connecting my interpretation of logical atomism to its origins in nineteenth-century mathematics and its paradigmatic expression in Principia Mathematica. And I criticized the traditional interpretation on which logical atomism is a kind of British empiricism. This view is so widely-held that it merits being called the general impression of logical atomism, which is why I criticized it at length.

Logical atomists, in my view, are term busters. Our practice of term busting is only viable as a philosophy given a logic that is sufficiently powerful. Otherwise, we as philosophers have little basis for accepting, much less for practicing, logical atomism. A logic that is capable of successfully giving a logical reconstruction of notions like number and class, and philosophical notions that are at least as complex, is required for a viable form of logical atomism.

The centrality of logic to logical atomism, on my interpretation of the view, accounts for its origins and for what we find in Russell’s writings. And it has an interesting application to Wittgenstein’s philosophy of logic. Trying to find logical atomism, as I understand it, in Wittgenstein’s works gives rise to an interesting narrative, which I partly gave in this work.

But the phrase ‘making logic central’ is vague. What has been sorely wanted for some time is an example of how a logical atomist would, in this day and age, philosophize. What does a modern philosophical work on a logical atomist’s ‘logic-first, logic-last’ philosophy look like? I tried to answer that question by giving an example. That example was this very work.

I am a logical atomist. I practiced logical atomist philosophizing in the second part of this work. I gave a philosophy of logic as pure logic, a formal system of Z-types that fits that account of logic, and then an ontology for pure logic consisting of completely general facts with no constituents and logical concepts. All of this admittedly leaves some problems unsolved. But these were not my focus here in answering my central ‘What is it?’ question.

My logical atomism that I develop in the second part is but a species of a broader genus. Clarifying the genus was my chief aim here. But I will defend my species of logical atomism, as I believe that it lays the foundation for rich and varying applications of the logical atomist’s term busting practice in philosophy. It is that foundation for my own philosophical work, and the new understanding of logical atomism that inspires it, that I will defend here today.

Professors, I am grateful for your attention to my work. I am prepared for your questions.

Bees Fly Towards Colors, Not Numbers

NPR recently published a piece titled, “Math Bee: Honeybees Seem To Understand The Notion Of Zero” that leads:

Honeybees understand that “nothing” can be “something” that has numerical meaning, showing that they have a primitive grasp of the concept of zero.

To which my response was, What? Sadly, the study is behind a paywall, so I had to rely on the description of the experiment given in the NPR article. And I am not an entomologist, so my ignorance of bees stands open to correction. So I am going to confine my argument to the following:

Premise Bees can detect and respond to different wavelengths of light (so as to behave differently when presented with different colors).

Premise Detecting and responding to different wavelengths of light is a more common biological capacity than detecting and responding to arithmetic features of an environment.

Conclusion The bees in the given experiment, as described in the NPR piece, could equally be, and indeed more likely are, responding to different colors and not grasping arithmetic features of their environment.

That qualifier in bold is important: perhaps the full experiment has further conditions that rules out the bees responding, speaking-loosely, to colors as opposed to numbers. But the experiment as described in the piece does not show at all, so far as I can tell (being massively ignorant about bees), that bees can understand zero, much less any arithmetic notion. Here is the experiment, as described in the piece:

…[They] lured bees to a wall where they were presented with two square cards. Each card had a different number of black symbols, such as dots or triangles. [They] trained one group of bees to understand that sugar water would always be located under the card with the least number of symbols…The bees quickly learned to fly to the card with the fewest symbols, an impressive feat…The researchers presented the bees with a card that had a single symbol — and a blank card that had nothing on it. The bees seemed to understand that “zero” was less than one, because they flew toward the blank card more often than you’d expect if they were choosing at random — although they weren’t that good at distinguishing between the two.

Here’s the experiment then:

  1. Bees are trained to fly towards cards with a non-zero number of symbols on them such that: the fewer symbols on the card indicates the reward.
  2. Bees are then confronted, for the first time, with a card with one symbol and a blank card no symbols such that: the card with no symbols has the better reward.

The results are interesting:

  • Bees do not randomly choose between cards, nor fly towards the card with one symbol, the one that they have been conditioned to fly towards: they fly towards the card with no symbols that they have not been conditioned towards at all.

The experimenters then infer that the best explanation for this is that the bees understand zero. But this is not the best explanation, so far as the description goes. Why not the following description of the experiment:

  1. Bees are trained to fly towards cards with less black coloring on them such that: the card with less black coloring indicates the reward.
  2. Bees are then confronted, for the first time, with a blank card with no black coloring and a card little black coloring such that: the blank card has the better reward.

The result is then still interesting, but more readily explained by their conditioning and by a trait that is more common, namely, color perception:

  • Bees do not randomly choose between cards, nor fly towards the card with one symbol, the one that they have been conditioned to fly towards: they fly towards the card with no symbols that they have been conditioned towards only by prior experience with other cards with different colors.

In other words, the hypothesis that bees have been trained to orient themselves away from the color black or towards another color more readily explains why they should move towards a blank card that they have never seen before. It is also a more common trait, which increases the odds that color-perception is what is really going on, not counting. The TL;DR version: don’t rule out more plausible alternatives by analogizing with your own introspective case. Prefer one that only requires visual capacities rather than arithmetic capacities.

Frege’s “Despondency” and Academic Writing

I was struck by a passage in Frege’s foreword to Volume 1 of the Grundgesetze der Arithmetik (Basic Laws of Arithmetic). I suspect also that I was only struck by the passage in light of my own experience with writing a dissertation, and so a book-length academic work, of my own.

Here is the context. In 1879, Frege published Begriffsschrift (Concept-Script). In 1884, Frege published Die Grundlagen der Arithmetik (The Foundations of Arithmetic). Loosely-speaking, these works may be viewed as formal and philosophical antecedents, respectively, for Frege’s Grundgesetze. But it was published in 1893, almost ten years later. Why did it take so long?

Frege goes on for a couple pages explaining how his new work has new primitive notions and how the passing years “have seen the work mature.” This is fairly typical: philosophers, especially honest and deeply thoughtful ones like Frege, rehash their views. Then Frege writes something that really struck me (page XI):

…I arrive at a second reason for the delay: the despondency that at times overcame me as a result of the cool reception, or rather, the lack of reception, by mathematicians of the writings mentioned above [Begriffsschrift andGrundlagen], and the unfavorable scientific currents against which my book [Grundgesetze] will have to struggle. The first impression alone can only be off-putting: strange signs, pages of nothing but alien formulae. Thus sometimes I concerned myself with other subjects. Yet as time passed, I simply could not contain these results of my thinking, which seemed to me valuable, locked up in my desk; and work expended always called for further work if it was not to be in vain. Thus the subject matter kept me captive. All that is left for me is to hope that someone may from the outset have sufficient confidence in the work to anticipate that his inner reward will be repayment enough, and will then publicize the results of a thorough examination.

A page later, Frege bluntly laments, “Otherwise [if I do not get such a reader], of course, the prospects for my book are dim.”

This cuts deep when I think about the reception that my dissertation is likely to receive, namely, none. Many individuals pour their work and closely-held beliefs into their dissertation. My uneducated guess is that most have yet to find a reader of the sort Frege sought. This can lead to melancholic thoughts, such as

Why am I spending my time doing this? Nobody will read it. It does not matter. I will not get a permanent academic position in which I can unpack the implications of this work anyway.

Thoughts like that can interfere with the writing. But the work keeps calling me back, as Frege’s called him. And Frege ends on a happy note that usually is enough to dispel such brooding moods (page XXV):

The distance [of my logical standpoint] from psychological logic seems to me to be as wide as the sky, so much so that there is no prospect that my book will have an effect on it immediately. My impression is that the tree that I have planted has to heave an incredible load of stone to make space and light for itself. Still, I will not give up all hope that my book will eventually aid the overthrow of psychological logic.