Just a Humble Philosopher
What are we trying to do when we teach philosophy? This question was broached recently at a conference I took part in on the role of formal methods in philosophy. It was maybe one of the most Very Extremely Online philosophy conferences the world has yet seen, being announced in advance on one blog, livetweeted by two different philosophers, with questions from twitter being asked at a roundtable at the end - which is now going to be blogged about here! (Just want to include a note of thanks to Samuel Fletcher, not only for his role in organising the conference but also for offering helpful feedback on this very blog post!) This isn’t the last time I am going to blog about the conference, but right now I want to focus on a rather curious element of the conference and reflect on what it means for those who engage with students of philosophy.
The conference had us all thinking about what we hope formal methods can do for various people who interact with the philosophy world. But first and foremost, it seemed to me, we had in our mind our undergraduate students, the people for whom we most directly act as educators, and with whom we share many conversations - yet who at the same time cannot for the most part reasonably be expected to go on to do philosophical research. Whatever it is formal methods are going to do for these people, it is pretty unlikely that it will be providing either an interesting object of future study or a method for carrying out research. What then do we hope they gain from their study of formal methods?
Now, of course, an obvious answer is: we think the theories and methods we teach are intrinsically interesting, and a good thing for anyone going through a liberal education to be aware of. Since the sorts of tools we typically have in mind here are the basics of logic, probability theory, and decision theory, there is some plausibility to that - these are (arguably!) not mere uninterpreted calculi but are basic elements of deep theories about what constitutes rationality, one of the core concepts of a philosophical education. But surprisingly, this was not that often raised at the conference! Likely, I guess, because it was presumed that everyone in the room was already persuaded of this and it would be of little profit to belabour the point. But, still, it was rare that people put forward the intrinsic gain from familiarity with these materials as the desired benefit that could accrue to our students from our teaching formal methods.
Instead, the answer which various people articulated one after the other was: an undergrad education in philosophy should aim to induce wonder and humility before the world’s complexities, and we think formal methods are one good way of doing that. (I stress now just to be clear: no claim of uniqueness on behalf of formal methods was made in this regard.) Here the idea was, broadly, that formal methods are good because they allow students to see clearly i) paradoxes that inescapably result from rigorously following through certain common assumptions taken together, ii) the limitations of what can actually be shown when one is precise about one’s assumptions and rules of inference, and iii) at more advanced levels anyway, just the raw difficulty of precise reasoning. Of course, this was music to my ears given my previous post so I was hardly inclined to disagree, but it did surprise me none the less!
My surprise came in particular from the contrast between this and one of the other main themes of the conference: the frequency with which our students have what people termed “math anxiety” (I reproduce their Americanism without passing judgement - edit: Rineke Verbrugge tells me on Facebook that there is actually some academic writing on the topic, going back to Sheila Tobias, so check it out!). This is the habit of our humanities students to think that symbolic reasoning is somehow intrinsically difficult and beyond their powers, and to feel especial fear and shame at the prospect of being seen not to be good at it, and thus displaying some hesitancy or avoidance about engaging with formal courses. Such math anxiety had been noted (and in some cases felt) by many of the teachers in the room, and all agreed that it was a problem which we needed to think seriously about as teachers - how can we help students out, lessen their fear, and help them get the most out of the resources we could make available to them?
Well, the more I have thought about it, the more I have been struck by this tension - there is a benefit to making students experience difficulty, limitation, the inevitability of failure. Let logic be their Kobayashi Maru! But in formal philosophy especially (but no doubt in other areas too) we have a real concern that students’ anxiety about difficulties doesn’t prevent them from being able to engage with and thus learn from from the material. The obvious way to do that is to give students puzzles they can handle, which can be somewhat contrary to the previous point.
There is no suggestion, of course, that this is an irresolvable tension. For one thing, one may simply try to design lessons so this happens in sequence - draw the students in by showing them soluble puzzles and guiding them very gently into it, then later making them confront harder limits. And, more generally, good teaching involves threading this needle: a sensitivity to the particular needs of the students before you, guiding them so they can see where the limitations are -- not as some failing of theirs but as a difficulty that comes out of a structural feature of the subject. But good teaching also involves ensuring that pupils are pushed to actually do as well as they can, to take on real challenges and meet them as such. We ourselves came to see the value in this difficulty, after all - it is learnable because we learned it, so we may be the ones who teach it in turn. But even if the tension is not inevitable, it is a tension. We want people to experience real difficulty, but we also want to assuage their fears (especially marked for mathematical courses) that this is just too difficult for them. The easiest way to respond to each desideratum is to do something that undermines one’s ability to fulfil the other.
But I wish to suggest a deeper underlying unity between the two desiderata than the mere fact that they can be jointly satisfied. I take it that the point is this: we wish students to have the experience of seeing themselves as limited not by personal failings, but by something like a real difficulty in the world. Philosophy begins in wonder, and teaches humility, but it is not the humility of someone with low self-regard, or wonder at how one can be such a wretch. We wish to put students in a place where they leave their degree realising that there is much to doubt about confident pronouncements made on behalf of what a Reasonable Person would do or what Rationally We Surely Must Believe. To do that in the way we intend, they must be able to recognise the puzzlement that arises at such things as not reflective of a mere failure on their own part to understand what is being said. It is rather reflective of just how slippery such notions turn out to be when one is careful, how much real cause there is for doubt on such matters. There is a certain kind of confidence we must teach (or encourage) in order to make people humble in the right kind of way.
Seen as such, I think helping students overcome their math anxiety can be connected to the broader goals of a philosophy degree. When students go out into the world they will encounter claims confidently put forward couched in statistical garb, and all sorts of claims from various parties to be logical where their opponents are not - I have even seen it said in a semi-popular venues recently that the correspondence theory of truth is the axis on which the present culture war turns. The fact that claims that use or reference logic or statistics play this peculiar social role adds a peculiar urgency to addressing math anxiety in particular. In so far as such claims activate the anxiety, students may find themselves unable to really engage with the claims' substance, and so unable to properly internalise them where that would be a good idea, or subject them to informed critique where that would be the better course of action. Math anxiety by its nature stands in the way of gaining a true understanding of one’s own capabilities and of the material we wish to teach. But it also is a peculiar barrier to understanding a class of socially significant utterances, claims about people or society which underpin normative arguments and policy suggestions. Formal philosophy teaching, if it is done well enough to overcome math anxiety, can play a vital role in producing informed future citizens - something I take to be a key part of the liberal arts educational ideal (see also).
But how exactly does it do that? Here we come back to the first desiderata, inducing wonder. As a community, we philosophers are far too fractious and in any case ill-trained to agree on some particular doctrine or theory that we should hope students will interpret claims through. But we do tend to retain some faith in something like the Socratic mission, that encountering such claims in an open-minded fashion is somehow good for oneself, and probably good for the polity too even where not appreciated as such. As the answers to the questions posed during round table discussion suggested to me, many of us think that when this is done, students will be in a state of wonder. Students who have overcome their math anxiety, who understand the formal tools we teach and their limitations, will be humbled. They will much less confident in many confidently made claims, much more likely to think claims about the world need to be thought through carefully, and perhaps not be in a position to say anything further until more study has been carried out. It just turns out that often inducing this kind of wonder requires overcoming some math anxiety - it requires one to see the difficulties one has in understanding what is being claimed as difficulties inherent in the subject matter rather than problems with the self.
There is an apparent tension between teaching humility and helping people overcome math anxiety, but on some level, this is to be resolved in a Hegelian fashion. The true humility, the sort of wonder which we wish to induce as philosophers, can only be achieved when one has achieved a certain degree of well-founded confidence in one’s ability to understand and assess claims. Many claims of interest are about or couched in logical or mathematical terms, and our tools are especially well suited to helping people recognise paradox and perplexity; formal philosophy hence has an important role to play in a philosophical education.
The conference had us all thinking about what we hope formal methods can do for various people who interact with the philosophy world. But first and foremost, it seemed to me, we had in our mind our undergraduate students, the people for whom we most directly act as educators, and with whom we share many conversations - yet who at the same time cannot for the most part reasonably be expected to go on to do philosophical research. Whatever it is formal methods are going to do for these people, it is pretty unlikely that it will be providing either an interesting object of future study or a method for carrying out research. What then do we hope they gain from their study of formal methods?
Now, of course, an obvious answer is: we think the theories and methods we teach are intrinsically interesting, and a good thing for anyone going through a liberal education to be aware of. Since the sorts of tools we typically have in mind here are the basics of logic, probability theory, and decision theory, there is some plausibility to that - these are (arguably!) not mere uninterpreted calculi but are basic elements of deep theories about what constitutes rationality, one of the core concepts of a philosophical education. But surprisingly, this was not that often raised at the conference! Likely, I guess, because it was presumed that everyone in the room was already persuaded of this and it would be of little profit to belabour the point. But, still, it was rare that people put forward the intrinsic gain from familiarity with these materials as the desired benefit that could accrue to our students from our teaching formal methods.
Instead, the answer which various people articulated one after the other was: an undergrad education in philosophy should aim to induce wonder and humility before the world’s complexities, and we think formal methods are one good way of doing that. (I stress now just to be clear: no claim of uniqueness on behalf of formal methods was made in this regard.) Here the idea was, broadly, that formal methods are good because they allow students to see clearly i) paradoxes that inescapably result from rigorously following through certain common assumptions taken together, ii) the limitations of what can actually be shown when one is precise about one’s assumptions and rules of inference, and iii) at more advanced levels anyway, just the raw difficulty of precise reasoning. Of course, this was music to my ears given my previous post so I was hardly inclined to disagree, but it did surprise me none the less!
My surprise came in particular from the contrast between this and one of the other main themes of the conference: the frequency with which our students have what people termed “math anxiety” (I reproduce their Americanism without passing judgement - edit: Rineke Verbrugge tells me on Facebook that there is actually some academic writing on the topic, going back to Sheila Tobias, so check it out!). This is the habit of our humanities students to think that symbolic reasoning is somehow intrinsically difficult and beyond their powers, and to feel especial fear and shame at the prospect of being seen not to be good at it, and thus displaying some hesitancy or avoidance about engaging with formal courses. Such math anxiety had been noted (and in some cases felt) by many of the teachers in the room, and all agreed that it was a problem which we needed to think seriously about as teachers - how can we help students out, lessen their fear, and help them get the most out of the resources we could make available to them?
Well, the more I have thought about it, the more I have been struck by this tension - there is a benefit to making students experience difficulty, limitation, the inevitability of failure. Let logic be their Kobayashi Maru! But in formal philosophy especially (but no doubt in other areas too) we have a real concern that students’ anxiety about difficulties doesn’t prevent them from being able to engage with and thus learn from from the material. The obvious way to do that is to give students puzzles they can handle, which can be somewhat contrary to the previous point.
There is no suggestion, of course, that this is an irresolvable tension. For one thing, one may simply try to design lessons so this happens in sequence - draw the students in by showing them soluble puzzles and guiding them very gently into it, then later making them confront harder limits. And, more generally, good teaching involves threading this needle: a sensitivity to the particular needs of the students before you, guiding them so they can see where the limitations are -- not as some failing of theirs but as a difficulty that comes out of a structural feature of the subject. But good teaching also involves ensuring that pupils are pushed to actually do as well as they can, to take on real challenges and meet them as such. We ourselves came to see the value in this difficulty, after all - it is learnable because we learned it, so we may be the ones who teach it in turn. But even if the tension is not inevitable, it is a tension. We want people to experience real difficulty, but we also want to assuage their fears (especially marked for mathematical courses) that this is just too difficult for them. The easiest way to respond to each desideratum is to do something that undermines one’s ability to fulfil the other.
But I wish to suggest a deeper underlying unity between the two desiderata than the mere fact that they can be jointly satisfied. I take it that the point is this: we wish students to have the experience of seeing themselves as limited not by personal failings, but by something like a real difficulty in the world. Philosophy begins in wonder, and teaches humility, but it is not the humility of someone with low self-regard, or wonder at how one can be such a wretch. We wish to put students in a place where they leave their degree realising that there is much to doubt about confident pronouncements made on behalf of what a Reasonable Person would do or what Rationally We Surely Must Believe. To do that in the way we intend, they must be able to recognise the puzzlement that arises at such things as not reflective of a mere failure on their own part to understand what is being said. It is rather reflective of just how slippery such notions turn out to be when one is careful, how much real cause there is for doubt on such matters. There is a certain kind of confidence we must teach (or encourage) in order to make people humble in the right kind of way.
Seen as such, I think helping students overcome their math anxiety can be connected to the broader goals of a philosophy degree. When students go out into the world they will encounter claims confidently put forward couched in statistical garb, and all sorts of claims from various parties to be logical where their opponents are not - I have even seen it said in a semi-popular venues recently that the correspondence theory of truth is the axis on which the present culture war turns. The fact that claims that use or reference logic or statistics play this peculiar social role adds a peculiar urgency to addressing math anxiety in particular. In so far as such claims activate the anxiety, students may find themselves unable to really engage with the claims' substance, and so unable to properly internalise them where that would be a good idea, or subject them to informed critique where that would be the better course of action. Math anxiety by its nature stands in the way of gaining a true understanding of one’s own capabilities and of the material we wish to teach. But it also is a peculiar barrier to understanding a class of socially significant utterances, claims about people or society which underpin normative arguments and policy suggestions. Formal philosophy teaching, if it is done well enough to overcome math anxiety, can play a vital role in producing informed future citizens - something I take to be a key part of the liberal arts educational ideal (see also).
But how exactly does it do that? Here we come back to the first desiderata, inducing wonder. As a community, we philosophers are far too fractious and in any case ill-trained to agree on some particular doctrine or theory that we should hope students will interpret claims through. But we do tend to retain some faith in something like the Socratic mission, that encountering such claims in an open-minded fashion is somehow good for oneself, and probably good for the polity too even where not appreciated as such. As the answers to the questions posed during round table discussion suggested to me, many of us think that when this is done, students will be in a state of wonder. Students who have overcome their math anxiety, who understand the formal tools we teach and their limitations, will be humbled. They will much less confident in many confidently made claims, much more likely to think claims about the world need to be thought through carefully, and perhaps not be in a position to say anything further until more study has been carried out. It just turns out that often inducing this kind of wonder requires overcoming some math anxiety - it requires one to see the difficulties one has in understanding what is being claimed as difficulties inherent in the subject matter rather than problems with the self.
There is an apparent tension between teaching humility and helping people overcome math anxiety, but on some level, this is to be resolved in a Hegelian fashion. The true humility, the sort of wonder which we wish to induce as philosophers, can only be achieved when one has achieved a certain degree of well-founded confidence in one’s ability to understand and assess claims. Many claims of interest are about or couched in logical or mathematical terms, and our tools are especially well suited to helping people recognise paradox and perplexity; formal philosophy hence has an important role to play in a philosophical education.
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