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I have very little experience in philosophy, so I am not sure if this question is common (I could not find anything on it). This site seemed to be most fitting for the question, but if this question does not fall under "philosophy," I will move it somewhere else.

I was wondering how we actually learn math and science (physics). Some people say that it is important to "understand" the formulas/equations. However, if anyone were asked what 5 divided by 5 is, they would immediately respond 1. Most people probably aren't picturing a group of 5 apples (for instance), making groups of 5 out of those apples, and counting how many groups they have. This suggests to me that we do not really understand math/physics -- it seems to me like it is memorization or pattern recognition.

Yet, if I asked someone how many groups of 5 apples they could make from 5 apples, they would know to do 5 divided by 5 and end up with 1. Here, it seems like the person has a true understanding of the concept of division.

Do we understand math/physics or just memorize it? How are we able to seemingly do both? Of course, the example I gave was very simplistic, but I can find many more. For example, I could ask someone with basic physics knowledge to calculate the power in a circuit given that there is 5 volts across it and 1 amp running through it. Most people will quickly calculate that there is 5 watts consumed without even thinking about the "meaning" of the equation they are using (power = voltage * current). Yet, if I asked them to explain to me why they used this formula, it would not be too hard for them to prove that it is correct. In other words, they have an understanding of the formula, but did not apply this understanding when asked to solve a simple problem.

This is just something that has been bothering me, and I was wondering if there was a logical explanation for it. Is our knowledge understanding or memorization? Is one form of knowledge more advantageous than the other? Why do it look like we have both forms? Thank you!

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    If people only memorized math equations then they would not be able to solve equations that they didn't already know the answer to. Imagine Alice who has never added 238492893482893 and 2308498234902903 together before. If she only had "memorized" the certain addition problems she's done before, she wouldn't know how to add those two numbers, but clearly we know that anyone who has learned addition can add them together. We know the rules work because we are shown proofs of them; when kids learn math teachers use tangible examples like apples because it helps the kids visualize the proofs. – Not_Here Jul 28 '17 at 14:50
  • But I am genuinely confused as to why you think the first example only shows that people memorize something and don't understand it. If you ask someone to divide two numbers they have never divided before and they do it perfectly, how could they have memorized the answer? Why does them not picturing groups of apples and instead just using a long division algorithm mean that they only memorized the problem and not truly understood the problem? – Not_Here Jul 28 '17 at 14:52
  • Chomsky's poverty of the stimulus argument is slightly related and might interest you. He argued against Skinner's behaviorism idea that language is developed by responding to external stimuli; Chomsky argues that if that were true people would not be able to understand novel sentences said to them that they had never encountered before. We understand patterns and how they work (patterns in language and math) and then we use those patterns when we solve problems. If it was just memorization we couldn't do anything new. – Not_Here Jul 28 '17 at 14:55
  • By "learn" vs. "memorize," are you asking whether physics explains or merely describes? – Geremia Jul 28 '17 at 18:24
  • This is a great question. I'm afraid formalists would be unable to answer what is the relation between 5 and 5 apples. Virtually all American mathematicians are trained in the formalist school. – George Chen Aug 28 '17 at 19:35
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Math and philosophy was in the acient Greece very connected and many times studied simultaneously as they where (and in some parts of the world still is) thought to coincide.

This is because math describes the fundamentals of nature, it models the world around us to a degree that we, as humans, find satisfying. In the same manner philosophy concerns answers that cannot be mathematically measured, because the subject being studied is a question of emotion, lack of it, reason and sow on.

Just because we are satisfied does not mean that math is completely true, math and philosophy are the only two non-empirical subjects in modern academia, which from a natural science perspective also means that it cant be proven and not to a full extend understood.

Within math it is easy to make proves that is limited by a "false math" the bunny and the turtle is a classic. The bunny can never pass the turtle because thenever the bunny reaches the point the turtle has just left, the turtle has moved on to a new point and sow on. In this limited math, this is true, even though we know it to be false because our math includes time.

We have accepted math to be true because it describes the world in a sensibly way, and it continues to do sow because math constantly develops. Matrices, Fourier transforms and quantum mechanics are examples of this. But we do not know, if all we know, is that the bunny cant pass the turtle.

To answer the question, we don't understand math or fore that matter anything, but we can model it daim well, and this is "proof" enough fore us to believe in it.

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Ok, maths in my opinion and in the opinion of others such as Plato and Pythagoras, is a very important part of Philosophy. It defines what is fact, it defines what is rational, and it defines what is true, and Philosophy is all about truth. For example, 1+1=2 is an absolute truth, which we usually learn and understand as a child. When it comes to memory... this is a different argument, there are many opinions on our memory. Some like Plato say we knew everything already before birth, but had forgotten in when born. If your talking about the behavioural side, I would say it is more about understanding, for example- algebra, I don't remember algebra from my maths lessons because I simply didn't care or really understand it- my memory therefore thinks its useless so I now know nothing about algebra. Whereas in religious education from high school, I remember the 10 commandments, this is because I understood they were important and the idea appealed to me because I found it interesting, therefore I have remembered it- because I understood the absolute moral code- e.g. do not kill. I hope some of this was useful, if I have misunderstood your question- apologies! -N

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Yes, arithmetic rules are memorized just like the rules of board games are memorized. The most important evidence is that everyone calculates in his native language ( or language of instruction) no matter how fluent they are in a foreign language.

Our understanding of the rules of chess is demonstrated by our ability to play chess; our understanding of arithmetic rules is demonstrated by our ability to solve math problems we've never seen before.

There are other sets of arithmetic rules that require a lot more memorization but offer the benefit of greater speed of calculation.

Arithmetic is sometimes applicable to practical problems, sometimes not. For most of us who learned arithmetic from school, where arithmetic applies is also memorized.

Our savage ancestors learned arithmetic by exactly the opposite means.

First, they conceived the number of things. E.g. two pebble, three pebbles, four pebbles, etc. Notice that there were always things or measure words following a numeral.

Then, through experiences, they noticed that two pebbles and two pebbles together are four pebbles.

Then some wise persons used such rules of thumbs as "two pebbles and two pebbles are four pebbles" to predict the outcome instead of putting two pebbles next to two other pebbles and count the total.

At this point, measure words or things following the numeral became too cumbersome and people sometimes dropped them for brevity's sake. Then they got such rules of thumbs as "two and two equals to four." The ancient Chinese mathematics book Nine Chapters was a rule-of-thumb era math book; it demonstrated this transition from number of things to number: every problem is a practical problem about land, length or weight, but the solutions simply dropped measure words where a reader would have no problem in understanding that two in this problem meant two dou (bushels), two in that problem meant two mu (acres).

As the number of rules of thumbs grew, someone figured out that some rules of thumbs can be deduced from other rules of thumbs and the number of rules of thumbs needed to memorize began to shrink. Finally, we got the arithmetic rule we have today.

I think, in order for children to have a deeper understanding of arithmetic, this part of our ancestor’s savage life might be worth reliving.

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