Mathochism: Types of reasoning

One woman’s attempt to revisit the math that plagued her in school. But can determination make up for 25 years of math neglect?

My second geometry class was yesterday, and the professor continues to impress me. I have yet to come up with a nickname for him — doing something with the fact that he’s Russian is tempting, but unfortunately, that is a language I don’t speak, though I have wanted to learn for years.

Anyway, we continued to study the foundations of geometry, and I realized I still didn’t quite understand the difference between inductive and deductive reasoning. So I asked him, and he explained it to me by saying that, in his experience, teaching more than 45 students at a time is untenable. His coming to that conclusion, he added, was an example of inductive reasoning, which is based on personal experience or experimentation.

However, if he had made that decision using numerous other people’s experiences, as well as the general assumption by the educational establishment that smaller class sizes lead to better student and teacher performance, then he was using deductive reasoning.

(I finally understand why so many Internet arguments end with people saying: “Your personal experience doesn’t make something a fact!” They’re clearly touting the superiority of deductive over inductive reasoning. At the same time, though, how many women have to say they hate being ordered to smile by strangers until we establish that as a fact?)

Deductive reasoning is the preferred form of reasoning for geometry, and I understand that, but I disagree that inductive reasoning is not useful and that intuitive reasoning is useless.

Let me amend that: I think people often think they’re using intuitive reasoning when in fact they are using inductive or even deductive reasoning. For example, we talk about getting a “gut feeling” when we meet someone, but a lot of the time, what we’re really doing is reading micro-expressions or analyzing actions that have proven to us in the past that the person who does this is usually pleasant or unpleasant.

Gavin de Becker talks about this in his book “The Gift of Fear.” He describes how a woman escaped a killer who repeatedly told her, very calmly, that he wasn’t going to hurt her, that he was just going to rob her. She noticed, without really registering it consciously, that his actions contradicted his words, and by doing the opposite of what he said — don’t move — she survived.

That is not intuitive thinking, which is not predicated on anything logical, but closer to inductive and deductive reasoning. And, of course, she was right.

At any rate, my geometry instructor doesn’t pooh-pooh intuitive or inductive reasoning. “We use it every day,” he told us. Using it in class, though, might be a mistake. Or at least, that is what I have deduced so far.

All text copyrighted by A.K. Whitney, and cannot be used without permission.



  • Interesting post. Philosophers have argued about the merit of inductive vs. deductive reasoning. Hume, for example, said that we cannot prove inductive reasoning (just because the sun rose every morning for all our life, that does not mean the sun will rise again tomorrow). In contrast, we can prove deductive reasoning using symbolic logic. But in almost any actual deduction, there are inductive premises. So deductive reasoning does not turn out to be on any more solid footing than inductive reasoning.

    Later, Quine argued in support of inductive reasoning by using evolution and natural selection. He said that we believe in inductive reasoning because if we did not we would not survive. So any animal that did not believe in inductive reasoning would not leave offspring.

    You might find these discussions interesting?

  • Well, exactly! It seems to me that deductive and inductive reasoning are not mutually exclusive. And that intuitive reasoning has a place too, since those “gut feelings” may be a manifestation of us picking up on tiny cues, whether its a drop in atmospheric pressure or a micro-expression.
    As for Quine, I see what he is saying, but as a child-free-by-choicer, I have to wonder where I fit in.
    Thanks so much for reading, and commenting!

  • I think when people say “your personal experience doesn’t make something a fact”, it is not so much that they are favouring deduction over induction, but rather that from their point of view, your experience represents only one data-point (and inductive reasoning is weaker the fewer points you have). However, from the point of view of the one reporting on personal experience, “experience” represents a lifetime of data-points.

    The other thing that might be happening is that they are approaching from a statistics point of view. Here, the reason “the plural of anecdote is not ‘data'” is because anecdotes fail to take into account the non-instances of connection (a classic example is the number of stories of “I thought of a person and then at that moment the phone rang, and it was them” – but you never hear the stories of “I thought of a person and the phone didn’t ring, or it did ring and it was someone completely different”).

    On the “being told to smile” thing, I think that there is only one witness to report on one’s emotional responses so saying “your experience doesn’t make it so” to “I don’t like it when people tell me to smile” is nonsensical. But to get to the general case “women tend not to like it when they are told to smile”, then you need lots of similar reports to establish the tendency (but if each individual report is rejected as “not being (necessarily) true” then each report gets treated as though it’s the first…)

  • An important point about math:
    one understands math inductively,
    but one proves math deductively.

  • Nicely succinct. Thank you Bill.

  • Yeah, and we shouldn’t discount the “some people like playing devil’s advocate” factor, or the “some people are just jerks” factor in many of these exchanges. Not that I expect to find such postulates at the back of my geometry book.