When I first got a job teaching algebra, I thought I would teach it by explaining exactly why the processes worked, and what principles they followed from. I soon found out that trying to do this made it more difficult, and that algebra was done better by one who didn’t think about what he was doing, but who simply executed operations more or less automatically. In fact, it was not clear that one could explain what they were doing, even in principle, when they did very basic algebraic things, like deriving the quadratic formula. What I called “thought” was an impediment to what is called “thought” in algebra and calculus.

Algebra and calculus, of course, go on to be the form of reasoning used in our understanding of the natural world. To the extent that they do, what a philosopher calls “thought” about nature is more and more marginalized and downplayed.

I am not sure that there is some possible synthesis of these two modes of thought.



  1. Peter said,

    December 12, 2009 at 12:11 pm

    Thinking is often an impediment to doing. Start thinking about what you are doing while you drive and you might get into an accident (say, by concentrating on the action of your foot and not on the car in front of you).

    We might say that thought slows you down. In doing so it is an impediment to gaining experience, which is most often what facility comes from whether in calculation or anything else.

    I’m not sure this adds anything to your line of thought, though.

  2. December 12, 2009 at 1:00 pm

    I’m reminded of Aristotle who deals with an objection “nature can’t act for an end, because natural things don’t deliberate” by saying “art does not deliberate”. He’s right. But it makes tings difficult when the art in question is one that is in the service of knowing, or when the art is taken as the very ideal paradigm of knowledge.

    • Peter said,

      December 12, 2009 at 1:53 pm

      Hehe, that was the very thing passage in Aristotle that I had in mind.

  3. December 12, 2009 at 1:28 pm

    Sure, you can execute an algorithm without any deep understanding of how it works; the same can be said for long-division as much as for algebra. But speaking as a mathematician (or, at least, as a person who graduated with a degree in mathematics) you can’t end there and be a mathematician.

    In short, I don’t buy your conclusion. What applies to beginning students of a subject isn’t necessarily the case of skilled practitioners.

  4. December 12, 2009 at 2:51 pm

    Then what is one doing when he, say, derives the quadratic formula, beyond moving letters around according to pre-set rules? At some point, algebra disconnected itself from geometrical or arithmetical quantity, and it cannot be joined to it again. Leaving this quantity behind, what is there? Just a bunch of rules that are followed like a dance.

    Here’s my claim: only two kinds of quantity exist: discrete and continuous (though I wouldn’t like to be pressed too hard on what “exists” means here) but very basic truths in algebra make no sense if one tries to explain them in terms of discrete or continuous quantity (there are problems enough with just “number lines” or “the real number system”, but even basic equations make no sense). So algebra is about something that doesn’t exist in quantitative terms. It’s not as much the study of an object as… well, I’m not sure. Better not to think about it and just do, which is what I think actually happens.

    To be clear, I think there are kinds of math that aren’t like this (Euclid, say) but Algebra and calculus are just not the same thing- and I don’t know how to relate the various kinds of math any more than I can relate algebra to philosophy.

  5. Robert said,

    December 12, 2009 at 3:17 pm

    I think Will’s point is that, in order to discover/derive/prove the operations of algebra and calculus, mathematicians had to think through the process by which numbers relate to one another and to things in reality.

    Now that we have a final working process, e.g., a quadratic equation or a derivative, we generally just use the operations themselves. It’s only in higher-level mathematics or in history of mathematics that the thinking and philosophy behind those operations needs to be spelled out.

    In a similar way, I don’t need to understand exactly the chemistry of combining eggs and yeast and flour and sugar and salt and the other ingredients in order to bake a cake; I just need to follow the recipe – unless I’m trying to work out a new recipe, which is a different thing from baking a cake.

  6. December 12, 2009 at 3:25 pm


    On my argument, your claim requires that higher level mathematics discover a quantity that is neither discrete or continuous. Such a thing does not exist.

  7. Brandon said,

    December 12, 2009 at 3:59 pm

    James Franklin has an interesting paper called “Aristotelian Realism” (PDF) in which he argues that modern mathematics is actually two distinct sciences: a science of quantity and a science of structure, and that the latter was largely a new discovery of the nineteenth century, one that grew out of considerations that were originally concerned with the former, and which can often shed light on problems to do with the latter, but actually has a different object (roughly, parts and their arrangements into wholes). I’m not sure how far this gets us, but certainly it’s worthwhile to think of the possibility that some of mathematics may have to do with quantity not directly but obliquely, e.g., with structures relevant to understanding quantities.

  8. December 12, 2009 at 4:02 pm

    Actually, Robert, I’m speaking of the actual process of deriving the quadratic formula, not the process of computing it for particular inputs.

    Consider a math problem my son was given this week:

    2*x + 10 = 20

    Dave knows how to follow the rules to solve this. I no longer remember the “rules” in a cookbook sense, but I understand how to solve it. I can make any change to both sides of the equation that leaves the two sides equal. For example, if I subtract the same number from both sides, that leaves the two sides still equal. If A and B are the same number, then A – 10 and B – 10 are clearly the same number. So I get

    2*x = 10

    Similarly, if I divide both sides by two, the two sides remain equal:

    x = 5

    I know I can do these steps because I know and understand the axioms of mathematics. It’s possible to solve the problem, as my son did, because he’s learned a pattern and is simply applying the pattern; but that’s not how I did it.

    With calculus, it gets more complicated, because no one differentiates or integrates the hard way. Everyone does it a few times, so that they know how it works; and then, as you say, they apply the rules. But again, the rules make sense for a reason, and you can use them better if you understand and think about the reason. Many don’t, I’m sure; but that doesn’t make following a cookbook recipe the essence of either algebra or calculus or any other higher math.

  9. December 12, 2009 at 4:04 pm

    James, I disagree about “quantities that are neither discrete or continuous”. In the example I gave above, and quite generally in calculus, the variables you’re dealing with might be unknown quantities, but they are understood to be continuous quantities. You pretty much always know what domain of numbers you’re dealing with.

  10. December 12, 2009 at 4:06 pm


    I’ve not read the paper you cite, but as you describe it, it makes sense to me. Linear algebra, for example, deals with vectors and matrices: collections of real quantities and variables, organized in particular ways. Complex numbers are a similar sort of thing. I.e., structure on top of quantity.

    • Ben Espen said,

      December 12, 2009 at 8:57 pm

      James Franklin has written quite a bit on Aristotelian philosophy of mathematics, although I took him to be claiming that structure [or relation] is primary in mathematics, and quantity subordinate.

      • Brandon said,

        December 12, 2009 at 9:54 pm

        It would depend, I imagine, on what one means by primary or subordinate.

        Given his argument in the post, I take it that James would be committed to drawing the lines somewhat differently from the way Franklin draws them, in any case.

  11. Mike said,

    December 12, 2009 at 9:27 pm

    I was about to respond and saw that Brandon has gotten there ahead of me. I took a masters in mathematics many years ago, and so far they have not made me give it back. I specialized in general topology, which is about as far from quantity as you can get and deals with structures and relationships, essentially: What do we mean when we say a is “near” to b? The later, when I went out in the big scary world I worked as an industrial statistician. Tests of significance can be boiled down to: when is a sample statistic ‘significantly different’ from another? I have sometimes said that general topology is “statistics without all those numbers.”

    So, I would concur that there are two kinds of mathematics: one that proves theorems and one that applies those theorems. I had started to say: pure versus applied mathematics; but maybe I should say: the science vs. the art of mathematics. You must think very hard when you “develop” the formulas; but shouldn’t have to think at all when you “do” them.

    You did not say in your post; but were you teaching algebra to math students or to physics/chem/etc. students? It might make a difference in the approach.

  12. Brandon said,

    December 12, 2009 at 10:17 pm


    I wonder if the puzzle may be resolved (or at least progress made toward resolving it) by making use of two distinctions. One is Mike’s suggestion that the distinction between art and science may be important here — it would make sense for it to be important for mathematics, since it certainly has some importance for logic. And this might be cross-cut with a distinction analogous to the one you’ve suggested for physics, between a first physics (dealing with fundamental conditions of change that any study of the world will be caught up in) and the sciences that follow after. Likewise, we could perhaps make sense of a first mathematics, dealing less with quantities than with quantity as such and its conditions, serving as the trunk, so to speak, for the various mathematical sciences. The combination of these distinctions, or any distinctions remotely like them, would seem to give a richer view of the relation between ‘thought’ and mathematics.

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