how we learn math

How many here believe that our mathematical skills can be nurtured without learning it from the environment? I don’t mean learning it from academic exercises or text book questions, I mean learning it from observations about how the things in the world work in accordance with mathematical principles. For example, do we learn that 2 + 2 = 4 because we’ve consistently experienced that whenever we have 2 things and we added them to 2 other things, we always get 4 things, or do we learn this solely based on the inherent meaning of ‘2’ and ‘+’ (such that we could do it all in our heads)?

One thing I’ve considered, if we go with the learning-from-the-environment perspective, is a hypothetical scenario in which one learns a different set of mathematical rules if his/her environment worked radically different that the one we take for granted. For example, suppose one developped in an environment made of nothing but water (this doesn’t have to be realistic - just as long as the principle of the argument is revealed by the scenario). Every time the individual takes 2 water droplettes and adds them to two other water droplettes, he/she gets not 4 but 1! Water droplettes congeal together when they come in contact with each other, so you just get a bigger water droplette. If this is consistently the environmental experiences this individual has, and no other sorts of experiences are had, would he/she end up understanding the rules of mathematics in a radically different way?

Well Id like to comment first on your analogy. The only problem I see with it is that yes, there is one droplet but there are two things to notice. The first is that the one droplet is larger than the four. In fact, its four times as large. Instead of equating that 2 things added to 2 things makes 4 things, they would conclude that 2 things added to 2 things makes something 4 times as big as any of the original things. I think saying this is exactly the same as saying 2+2=4.

The second is that you’re assuming you must mix the droplets. Say I have a box drawn on a table. I say the box has zero droplets in it. I then drop one in. I then drop another, but not on top of the first but to the side of it. I then add two more. Thus adding the two droplets to the original two. I think they would then say that there are four droplets and conclude 2+2=4.

Anyway, I think it was just a poor analogy. I think your idea is still a valid question. I just think its impossible to think of another way for it to be much like its impossible for you to imagine another physical dimension besides length, width, and height. You cant do it but that doesnt make it not possible to exist.

I dont think one could learn or understand the concept of math without real world experience. Im sure they could simply be taught the rules just like we do to a computer. But I dont think they would really “understand” it in the sense we do. I dont see how someone could be isolated for this to happen though. To have no natural exposure to math would seem to require you to not exist. I think math is a human invention used to understand nature. Its just very apparent at its basic level. You cant help but notice and see that if you have something in your hand, and you place another of that something in your hand, you now have a different situation in your hand. We call that different situation “more” of that thing. It is “more” in what we give the name “four” I dont think it really has any real meaning. Its just an efficient way for us to explain ideas and concepts and nature to each other.

Only if he/she could not differentiate the quantities of molecules in each individual droplet as opposed to the two droplets combined. The final form of the one droplet, after the two are combined, is a macroscopic scale, or duplicate, of each of the two droplets…but the difference isn’t in quanitity, but mass or size.

Every object is composed of smaller particular objects which can be counted as individuals. Even though the larger droplet looks like a smaller droplet, it cannot be counted as an equivalent number because it consists of a greater quantity of particular parts.

Verily, this is what I believe.

mathematics are a purely human construction of definitions that are (tentatively) applied to the world. they (numerals) never appear themselves in the world…

a “tree” appears in “nature”…

where in nature is it designated “tree”?

the same holds true for numbers

-Imp

I think Rapt0rzzz has it right with this quote. My analogy was a poor one, but for the life of me, I can’t imagine a better scenario. The fact of the matter is, mathematics describes the workings of our physical universe. So to imagine a scenario in which our experiences would lead us to adopt a different set of mathematical rules would be to imagine a completely different kind of universe, one that is impossible to conceive.

But I do think the analogy, as flawed as it is, does drive my point home: do we understand the rules of math because of our experiences with the way nature works, or is it something innate - or at least, independent from experience?

“Platonic” a priori references to numbers might be possible, that is, the formal “having-of-a-theory” is distinct from a theory itself, and is a necessary abstraction from experience, but the “truths” of natural numbers need not be conclusive in order to have a functioning mathematical theory. This is demonstrated in our physical axioms without our having, as Hawking put it, a “grand unified theory” of the universe.

Anyway, what I say is amatuer compared to these fellas. Have a look:

en.wikipedia.org/wiki/Edmund_Husserl

en.wikipedia.org/wiki/Gottlob_Frege

Impossible. You cannot experience divisions unless they are present objectively in the world. Regardless of the representation of numbers in language, the experience of quantity is absolute and determined by degrees of sensual stimulus. This is beyond language and has nothing to do with tacit agreement, which you claim is the “invention” of mathematics.

You contradict yourself when you say “to” the world. What is the world if not a mathematical construct?

Definitions represent content and substance, they signify experienced forms and objects. These objects do not depend on your “conceptualization” of them.

I would say mathematics is not known a priori, but it may appear so since we almost immediately experience it in nature. If you could, for example, take a human the second he/she was born and put them in a universe different from ours, they would learn the mathematic principles present there. I dont think they would not understand math in this new universe. I think they would learn it. However I dont think this is possible since a universe of this kind would have to be so different in physical structure (which we cant even imagine) that we couldnt live or function in it, never mind be capable of attempting to learn something there. But theoretically, I think we could learn the math. So I think its based on experience and we derive knowledge of math a posterior.

yes, they do. nothing exists besides “conceptionalization”…

-Imp

At such a point where this experience occured, it would be objectively true that “your half was bigger than my half.” This would be true for you and me. If, on the other hand, a different physiology did not permit me to experience the object as you do, the tacit agreement in language would not exist, and the question would not be a matter of “which half is bigger” but of “how does the object appear to me as opposed to you?” The answer would still be objective, because it would be true for you and me.

No, the absolute is proved in the experience of the object. It is directly real in its being, despite how it is described. Description with language is not the same thing as perception and impression of sensory data.

These are not real questions because the subjects which experience them are different, and cannot yield the same result in experience, much less experience the same attributes of an object’s beings. The problem is not “proving to me” a state of experienced affairs, but rather convincing yourself that you are not having them. You cannot do this, and you do not need “language” to do so.

Hardly. You do not need to know what the term “two” means to notice a spacial difference between an object “here” and an object “there” when in the proximity of both. Try again.

Now you are saying that a subject experiences a subject. This is semantics, because the experience of anything implies an object and a perception. Call it what you want, there is still a dichotomy of entities.

Definitions are attempts to establish agreement concerning the nature of an object. The object does not require a definition to exist. “Being” is absolute and without description in language. Or rather, failing to find a definition does not compromise the existence of an object.

Okay, Berkeley, whatever you say.

no, I am not the bishop…

-Imp

Okay Imp. You win.

[ahem]

Waiter!

Check please.