[cool_shit]

Terrible advice! I tell all of my students to find as many analogies as they can. You can never lose by increasing the number of ways in which you understand something. Analogies, some may argue, are at the heart of mathematics.

> Mathematics is the art of giving the same name to different things

Poincare.

> The art of doing mathematics consists in finding that special case which contains all the germs of generality

Hilbert.

> The vast majority of us imagine ourselves as like literature people or math people. But the truth is that the massive processor known as the human brain is neither a literature organ or a math organ. It is both and more.

John Green.

> Sometimes I think that creativity is a matter of seeing, or stumbling over, unobvious similarities between things - like composing a fresh metaphor, but on a more complex scale.

David Mitchell.

There is an ounce of truth in what you say -- metaphors can be abused to draw false conclusions. This does not mean one should cower from using them.

[ankurdhama]

Everyone has their own philosophy of mathematics. To me, the core idea is the idea of representations, such that those representations allow you to extract patterns (aka similarity or commonality) from various cases.

[nicklaf]

I can't help but indulge a bit and now ask whether or not we can't​ extract a common pattern from the two representations of mathematical cognition offered by yourself and GP. :-)

[ankurdhama]

Not here but may be on "meta hacker news".

[fmoralesc]

Aren't metaphors and analogies representations that allow us to extract patterns?

[ankurdhama]

To me, metaphors and analogies are about trying to learn some new concept(s) in terms of already known and familiar concepts.

[nicklaf]

As an opposing view, I offer you the following analogy, by mathematician Charles C. Pugh, which he draws between conceptions of metaphor as understood in natural versus mathematical language:

From "Real Mathematical Analysis" 1st edition, p. 9:

Metaphor and Analogy

In high school English, you are taught that a metaphor is a figure of speech in which one idea or word is substituted for another to suggest a likeness or similarity. This can occur very simply as in "The ship plows the sea." Or it can be less direct, as in "his lawyers dropped the ball." What gives a metaphor its power and pleasure are the secondary suggestions of similarity. Not only did the lawyers make a mistake, but it was their own fault, and, like an athlete who has dropped a ball, they could not follow through with their next legal action. A secondary implication is that their enterprise was just a game.

Often a metaphor associates something abstract to something concrete, as "Life is a journey." The preservation of inference from the concrete to the abstract in this metaphor suggests that like a journey, life has a beginning and an end, it progresses in one direction, it may have stops and detours, ups and downs, etc. The beauty of a metaphor is that hidden in a simple sentence like "Life is a journey" lurk a great many parallels, waiting to be uncovered by the thoughtful mind.

Metaphorical thinking pervades mathematics to a remarkable degree. It is often reflected in the language mathematics choose to define new concepts. In his construction of the system of real numbers, Dedekind could have referred to A|B as a "type-two, order preserving equivalence class", or worse, whereas "cut" is the right metaphor. It corresponds closely to one's physical intuition about the real line. See Figure 3. In his book, Where Mathematics Comes From, George Lakoff gives a comprehensive view of metaphor in mathematics.

An analogy is a shallow form of metaphor. It just asserts that two things are similar. Although simple, analogies can be a great help in accepting abstract concepts. When you travel from home to school, at first you are closer to home, and then you are closer to school. Somewhere there is a halfway stage in your journey. You know this, long before you study mathematics. So when a curve connects two points in a metric space (Chapter 2), you should expect that as a point "travels along the curve," somewhere it will be equidistant between the curve's endpoints. Reasoning by analogy is also referred to as "intuitive reasoning."

Moral: Try to translate what you know of the real world to guess what is true in mathematics.