Nor do I think [abstract ideas] are a whit more needfull for enlargment of Knowlege, than for Communication. For tho' it be a point much insisted on in the Schools, that all Knowlege is about Universals, yet I can by no means See the necessity of this Doctrine. It is acknowledg'd that nothing has a fairer title to the Name of Knowlege or Science than Geometry. Now I appeal to any mans thoughts, whether upon the entrance into that Study, the first thing to be done is to try to conceive a Circle that is neither great nor small, nor of any determinate Radius, so to make Ideas of Triangles & Parallelograms, that are neither Rectangular nor Obliquangular &c? It is one thing for a Proposition to be universally true, and another for it to be about Universal natures or notions. Because that The three angles of a Triangle are equal to two right ones is granted to be a Proposition universally true, it will not therefore follow, that we are to understand it of universal Triangles, or universal angles. It will suffice that it be true of the particular angles of any particular Triangle whatsoever.George Berkeley, Manuscript Introduction (c. 1708), folio 14, crossed out text omitted
It seems to me that the phenomenology of mathematics—that is, the nature of the subjective experience of mathematical practice and discovery—has played a crucial role in the Western philosophical tradition that has usually not been adequately appreciated by historians. One particularly important aspect of this phenomenology is that those who have spent a great deal of time engaged in mathematical practice often report the experience of the mind encountering an object. When learning a new branch of mathematics, they often speak of becoming acquainted with its objects. The mathematician manipulates these objects with her mind, and gets to know them. Sometimes the objects are experienced as offering resistance to certain directions of thought, sometimes as cooperating.
One can find descriptions of this kind of mathematical phenomenology in philosophermathematicians from Plato to Descartes to Gödel. (There is some fascinating discussion of Gödel's remarks on these matters in Penelope Maddy's excellent little book, Defending the Axioms.) The analogy between the mind's experience of encountering the idea of God and the mind's experience of encountering the idea of a triangle plays a crucial role in Descartes's arguments for the existence of God, and indeed there is reason to suspect that Descartes's entire theory of true and immutable natures stems from his experiential understanding of mathematical practice.
Kant, too, thinks that it is clear that, in mathematics, the mind encounters objects. Kant thought that, in order for our knowledge of these objects to be a priori, we must be responsible for constructing them. He then set out to explain why the mind experiences resistance from these objects, why we find that we cannot simply construct them however we will. For Kant, as for Descartes, the phenomenology of mathematics lies at the root of a much larger theory of a priori knowledge.
I was reading again through Berkeley's Manuscript Introduction today, and I was struck by an interesting feature of his argument against the use of abstract ideas in the philosophy of math, which I think can be found in a number of other places in his writings: Berkeley here privileges the mathematical experience of the novice over that of the expert. The novice sees the particular triangle diagram on the particular page, and does the Euclidean construction that shows that its interior angles add up to 180 degrees. By consideration of how he arrived at that result, he can recognize that the result applies, necessarily, to all triangles. But what he doesn't do is "tack[] together numberless inconsistencys" (folio 14, verso) to create an idea of a triangle "that [is] neither Rectangular nor Obliquangular &c," or "a Circle that is neither great nor small, nor of any determinate Radius" (folio 14). The mathematical expert, who effortlessly sees the universality of the result, comes to think that her proof is about some mysterious entity, The Universal Triangle. The novice, by contrast, correctly recognizes that the proof and the resulting theorem are about all the particular triangles.
Berkeley's approach to mathematical practice and mathematical phenomenology is reflective of his broader philosophical approach. For instance, a crucial point in Berkeley's theory of vision is the claim that normal adults interpret our visual experience so effortlessly that we are unaware there is any interpretation going on at all—we wrongly believe that relations of distance (for instance) are part of the content of vision itself, when in fact they are part of our interpretation. Similarly, the broader error of thinking that our words stand for abstract ideas stems from the effortlessness with which we apply them generally or universally. However, in order to understand the principles of human knowledge—that is, its basic and original materials—we need to disinterpret our experience and see the world like an infant—or, to use an example much beloved by Berkeley and his contemporaries, a man born blind and newly made to see. In the same way that the infant, or the newlysighted person, who has not yet developed the habit of interpreting vision, is an authority on what is really contained in vision prior to interpretation, so also the mathematical novice is an authority on the principles, the original materials, of mathematical knowledge.
To use another analogy to which Berkeley frequently appeals: it is precisely because of the effortlessness with which we speak our native language that it is difficult for us to gain explicit awareness of its grammar. One is much more aware of rules and sounds when learning a new language, but in one's native language one is aware only of the meaning. In the same way, it is the effortlessness with which the expert mathematician engages in mathematical practice that prevents her from recognizing the true nature of that practice, according to Berkeley. To understand how mathematical practice really works, one must return to the mindset of the novice struggling to derive any meaning from the lines and curves on the page. As in Berkeley's theory of vision, this is a process of disinterpretation.
It is also worth noting that the process of disinterpretation pursued in Berkeley's philosophy is essentially, in Kantian terms, the recovery of the unsynthesised manifold. The impossibility of Berkeleian philosophy is thus a fundamental principle of Kantian philosophy. And the question of which philosopher gives the correct account of the experience of the mathematical novice turns out to be absolutely crucial to the disagreement between them.
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"Berkeley here privileges the mathematical experience of the novice over that of the expert. The novice sees the particular triangle diagram on the particular page, and does the Euclidean construction that shows that its interior angles add up to 180 degrees. By consideration of how he arrived at that result, he can recognize that the result applies, necessarily, to all triangles."
This is an interesting take! And I believe the right one. Berkeley set out to show that there were no difficulties in knowledge where philosophers, like Locke, thought there were. A novice can understand perfectly well the reference of 'triangle' without the purported difficulty of abstracting it from particular triangles.
At the International Berkeley Society APA this past month, one issue that did come up is Berkeley's common sense approach to philosophy. Is the appeal to phenomenology suggestive of the fact that this approach was extended to mathematics, too?
You mention Maddy's Defending the Axioms. One of the issues discussed there is about whether philosophers (novices in comparison) can recommend revisionary practices to mathematicians (or experts) or supplement their accounts of justification or evidence (pp. 52, 53). Was his insistence on phenomenology or the experience of the novice the rejection of revisionary programs in mathematics or was this in fact a revisionary program itself (but from a noviceperspective)?
Some Berkeleyan scholars think that Berkeley was pursuing New Foundations for Geometry and Arithmetic. Now we might ask: is infinite divisibility of finite lines the common sense view or is its denial the common sense one? I think that a novice hearing that motion is impossible if finite lines are infinitely divisible, might reject infinite divisibility. There are places where Berkeley relies on phenomenology to reach the conclusion that finite lines are not infinite divisible. Would relying on the novice here or phenomenology be too costly?
All in all, this raises lots of interesting questions! I like the Kantian connection, too.
Posted by: David Mwakima at February 4, 2021 6:48 PMHi David,
Yes, I do think this goes together with Berkeley's denying that problems about knowledge exists (rather than solving them). And I do think that Berkeley ultimately thinks philosophy will recommend revisions in mathematics. Indeed, Berkeley's work on the foundations of calculus looks like this. But I don't think it's that the novice can criticize the experts. I think, rather, that the person who is new to mathematics is in a better position than the expert to give an actual step by step account of how the practice proceeds. By identifying how the practice proceeds in the good cases (like Euclidean geometry) we can get a better sense of how math is supposed to work, and thereby better recognize when something has gone awry. (Of course, according to Berkeley, something has gone awry in calculus.)
Posted by: Kenny Pearce at February 4, 2021 7:08 PM