Reconocer el trabajo del equipo

In this section I will return to the mathematical realm and explore how knowledge-how can arise here too. Agreeing with the anti-intellectualist claim that there is a substantial difference between knowledge-how and knowledge-that, and the broader picture from Wiggins on which the two kinds of knowledge are nonetheless strongly interrelated, we now have some ideas to bring back to apply to mathematics.10

To begin, let us connect Rav’s ideas to the epistemological literature. Rav claimed that the interesting mathematical knowledge is knowledge of proofs, since this is where the methods, techniques, concepts and interesting ideas, which are the essence of mathematics, actually reside. The thought then is that Rav’s big insight, put in the epistemological framework, is that the interesting knowledge in mathematics isknowledge-how. On this view, the knowledge that mathematicians are after is knowledge of how to solve problems,how to prove theorems, how to analyse data etc.

However, while I thought that Rav’s argument was sufficient to show that knowledge-how is of serious interest in mathematics in its own right, this is not to the exclusion of knowledge of mathematical truths and propositions. Instead the two are closely connected, in that mathematical knowledge-that of truths and knowledge-how of methods, techniques and strategies, are not easily pulled apart in practice. What do I mean by this exactly? Well, the thought is that each would be severely diminished without the other,

10_{Patrick Greenough has commented on this section that it should be noted that there}

is a coherent position for a kind of Ravian intellectualist, where the focus is on practical modes of presentation for mathematical knowledge-how as a kind of propositional knowl- edge. This doesn’t seem to be compatible with Rav’s own views due to the other material he has against Formalist-Reductionist claims, standing against there being underlying propositions for proofs or knowledge of them. However, there is room for this position in the debate and it does appear to be open for proponents of intellectualism to explore, though I shall not be doing so here.

both in the history and practice of mathematics. I want to now spend some time on various examples of how the distinction between knowledge-how and knowledge-that will be reflected in mathematics, as well as where we can see the interaction between the two taking place.

For one thing, a focus on knowledge-how in mathematics will contribute to a better picture of mathematics education. Work in maths education has shown a significant awareness of the need to teach students practical knowl- edge and skills. A prime example is found in the work of Gila Hanna who has over 30 years examined the importance of proof and proving in mathe- matics education, such as in (Hanna 1989), (Hanna & Jahnke 1996), (Hanna & Barbeau 2008) and (Hanna 2014).11 In fact, in (Hanna & Barbeau 2008), they engage with the Ravian view of proof and weigh up its importance for mathematics education:

We argue that what is true of mathematics itself may well be true of mathematics education: in other words, that proofs could be accorded a major role in the secondary-school classroom pre- cisely because of their potential to convey to students important elements of mathematical elements such as strategies and meth- ods. (Hanna & Barbeau 2008, p. 352)

They make the argument for this around two case studies of the benefits of teaching particular mathematical proofs, one of the quadratic formula (which I will discuss separately shortly) and one concerning angles inscribed in circles. The point is that learning proofs is an important way of also learning strategies that take us beyond merely learning the truth of the theorem, as well as allowing us to come to a more rounded understanding of mathematics.

There is a major Rylean point here about the process of learning: that learning a subject is often about being inducted into the practice of that subject rather than merely learning the truths associated with it. Ryle even includes mathematics as an example of this phenomenon:

The fact that mathematics, philosophy, tactics, scientific method and literary style cannot be imparted but only inculcated re- veals that these too are not bodies of information but branches of knowledge-how. They are not sciences but (in the old sense)

disciplines. The experts in them cannot tell us what they know, they can only show what they know by operating with clever- ness, skill, elegance or taste. The advance of knowledge does not consist only in the accumulation of discovered truths, but also and chiefly in the cumulative mastery of methods. (Ryle 1946, p. 15)

Actually, Ryle goes on further to suggest that knowledge-that rests on prior knowledge-how, as both discovery and deployment of our proposi- tional knowledge requires practical knowledge of how to discover and where the knowledge fits into the wider framework.

Effective possession of a piece of knowledge-that involves know- ing how to use that knowledge, when required, for the solution of other theoretical or practical problems. (Ryle 1946, p. 16)

The case for this in mathematics is particularly strong, as even understand- ing the language of mathematics is about knowing what can be done with the various concepts deployed. For instance, as one of the first things chil- dren learn, the ‘+’ symbol is directly associated with learning the process of adding numbers together and it is hard to imagine understanding what it means independently. Of course, the inculcation into mathematical prac- tice is not a one-off event, but a continuing development of knowledge of mathematics, both practical and propositional.12 There are some skills, abilities and pieces of know-how which are more general and others which are topic-specific, but this does not affect the point that mathematics in- volves the cumulative mastery of methods as well as knowledge of theorems and propositional statements.

That learning mathematics involves being inducted into practices, prac- tices which involve both knowledge-how and shared items of propositional knowledge, has clear impact on the claims from the previous chapter con- cerning mathematical concepts. Something which I quietly avoided flagging up earlier was that Rav’s idea of what proofs give us knowledge of (besides methods, skills, interactions and systematisations) included knowledge of

12_{As well as much more besides: being inducted into the practices of mathematics}

involves all kinds of additional learning, such as how to behave at conferences; how to present proofs on a blackboard; which journals to send which papers to; which math- ematicians are helpful, rigorous, friendly, quick at responding to emails; which funding bodies to apply for grants from etc.

mathematical concepts. There is room here for a strong stance on the na- ture of concepts, where concept possession is about the ability to make use of them in various ways, such as in distinguishing whether some object be- longs in the extension or anti-extension, deploying them in inferential moves, describing them correctly in appropriate linguistic settings, recognising the relations between them and other concepts etc. Such a move would bring concepts and knowledge-how close together, via the tight link between abil- ities and knowledge-how. I refrain from leaping into this discussion fully, but it certainly does not strike me as implausible in the mathematics case. In particular, the prominent place of informal concepts, open-texture and domain-specific reasoning in mathematics is suggestive of the idea that com- ing to know how to do mathematics involves coming to a tacit understanding of the sort of activities which are acceptable to carry out. I shall return to this point later in section 4.6.

Thus far, we have seen that on the picture I am presenting, mathematical knowledge-how is frequently prior to propositional mathematical knowledge. However, I also want to demonstrate that the Wigginsian observation is in full effect and the relationship goes the other way too, such that knowledge- that and knowledge-how are interdependent. Following Hanna & Barbeau, consider the example of the quadratic formula, i.e. that the solutions to an equation of the formax2+bx+c= 0 are given byx= (−b±√b2_−4ac)/2a. The point Hanna & Barbeau make is that learning the proof that the quadratic formula will always deliver the roots of a quadratic equation can teach a student the skills involved for several related techniques, such as the “completing the square” method and applications to examples beyond quadratics, such as quartic equations of particular forms. But, as a parallel point, the propositional knowledge of the truth of the theorem does deliver the knowledge of an easy way to solve a whole class of problems, one which a struggling student can perform almost mechanically even if they don’t un- derstand the reason that it works. That student can now know how to solve more quadratic problems than they did before. Indeed, if their difficulty is localised to just quadratics, that student might even be able to solve much more complex problems that require solving quadratics as a part. Obtaining the propositional knowledge of the quadratic formula acts as a key to unlock further knowledge-how.

categories. These are relations between categories that show them to be “es- sentially the same” in the case of equivalence, or equivalent to the “opposite” in the case of duality. The power of such results is immense in their ability to bring out connections between seemingly disparate areas of mathematics and to transfer theorems easily from one to the other without a fresh proof. This holds in a very strong sense, as Mac Lane puts it:

For more complicated theorems, the duality principle is a handy way to have (at once) the dual theorem. No proof of the dual theorem need be given. We usually even leave the formulation of the dual theorem to the reader. (Mac Lane 1998, p. 32)

In general, category theory thrives on these kind of links, and there are a large number of theorems about duality between categories. For example,

Stone’s representation theorem gives an isomorphism between Boolean al- gebras and certain topologies on sets (in particular: a topology on the set of ultrafilters of the Boolean algebra) and Birkhoff ’s representation theo- rem does the same for distributive lattices and partial orders. Generalis- ing, Stone duality refers to the broader class of categorical dualities hold- ing between topologies and partially-ordered sets, which allows us to move between different disciplines while straightforwardly transferring theorems. The philosophical significance here is that there is once again the lock-and- key phenomenon going on of knowledge-that providing the means to open a whole new range of methods and puzzle-solving techniques. While certainly it requires some background to establish dualities, the interesting mathe- matics lies not necessarily in the proofs or the methods used in the proof, but rather in the fact that the establishing of the representation theorems al- lows us to think about certain structures in two distinct but equivalent ways. The usefulness of this is emphasised by Abramsky as a ‘creative ambiguity’:

Mathematically, this distinction can be related to the duality between points and properties, in the sense of Stone-type du- alities: the duality between the points of a topological space, and its basic “observable properties”—the open sets. The par- ticular feature of domains which allows this creative ambiguity between points and properties to be used so freely without in- curring any significant conceptual confusions or overheads is that

basic points and basic properties (or observations) are essentially the same things. (Abramsky 2008, p. 494)

The Ravian picture, on which the theorem is the ‘headline’ to go with the interesting parts of mathematics which are embedded in the proofs, falls short on the example of representation theorems and Stone duality, in that the interesting mathematics does not reside in the relatively mundane proofs of the theorems, but instead in the new connections one can draw once the theorem is in place and the Gestalt-shifting in viewing well-known structures in entirely different ways. Knowledge of how to prove the theorems is an important discovery that establishes the truth of the duality and gives us knowledge thereof, but it is the latter knowledge of the truth of the theorems which is primary in opening up the new connections which can subsequently be drawn. The knowledge that is discovered about the vast network of con- nections between different mathematical structures is interesting and might well be entirely propositional. The propositional knowledge of these con- nections then opens up the scope for a whole range of additional methods, techniques and results, once again supporting the idea that the mathemat- ical knowledge is best understood in terms of interconnected propositional and practical knowledge.