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In high school nowadays most mathematics you learn is fairly 'old'. You have your geometry, all of which (taught in high school) was known to the Greeks more than 2 thousand years ago. You have medieval trigonometry and algebra. You also have 17th century calculus, and that's about it for most high schools.

Now for somebody seriously interested in mathematics, it is hard to decide where to continue your studies after completeling the high school curriculum. Of course you could just start reading about a certain field you are particularly interested in and disregard the other fields as much as you can, but this might give you gaps in your knowledge. You could just learn more advanced calculus, like people usually do, but you'll get the same dilemma after finishing multivariable calculus.

So what I've been doing is reading about the history of mathematics and just reading the major developments in chronological order. Pre-Greeks there was some mathematics, but almost all of it is extremely trivial for a high schooler, and there are no proofs, so the Greeks is where I started. What I'd then do is just read some of their works very selectively, just the most revolutionary stuff. For example:

  • Euclid's proof of the infinitude of primes
  • Proof of existence of irrational numbers
  • Archimedes proof of the area of a sphere

These are just some of the things off the top of my head, usually not taught in high school, which come to mind. And you keep progressing in time, quickly reaching the 17th century . My question is; would it be smart and logical to approach mathematics in this way? As an aspiring physicist, the 17th century is just where it begins to get complicated and I was wondering if it would be smart to keep on doing it like this up until around the 20th century. In my opinion, it flavours up the learning process, it is much more fun than reading a boring text book, it also shows the motivation, and of course reading from the original author themselves in some cases is beneficial (and if it isn't, there are a billion explanations online).

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  • $\begingroup$ Perhaps related is this earlier question about history and reading the masters. $\endgroup$ – Old John Dec 7 '13 at 23:08
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    $\begingroup$ I think it would be terribly complicated to take this approach seriously. Many results were not truly understood until decades if not centuries after their first appearance, for example. The history of calculus is full of erroneous methods for adding series, for computing derivatives, for estimating areas. There are also correct results whose modern proofs are much better in many respects than the original ones. I think studying historical sources is invaluable, but can only be appreciated once you have a good working understanding of modern issues. $\endgroup$ – Andrés E. Caicedo Dec 7 '13 at 23:10
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    $\begingroup$ @Phaptitude It is not always trivial. And even when some of the original approaches are not formal, and you know of a possible formalization, to identify the problems may be very hard. An example is the notion of "adequality" that Fermat used. Nowadays we use limits, but reading adequality as a limit computation is incorrect. $\endgroup$ – Andrés E. Caicedo Dec 7 '13 at 23:17
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    $\begingroup$ @Phaptitude Another example is the development of the theories of the continuum. There were long debates in the late XIX centuries over whether the geometric continuum is truly a set of points, or something else. It is hard to appreciate the issues without a good understanding of the foundations of analysis. And knowing the foundations is not enough to fully see where the philosophy ends and where the difficulties are mathematical. $\endgroup$ – Andrés E. Caicedo Dec 7 '13 at 23:20
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    $\begingroup$ It makes no sense to read, say, Newton as he did noting, and his "proofs" were no proofs at all, the same applies to all these greeks and italians and whatever. just read bourbaki, logic must take precedence over history. otherwise you will know something about history but nothing about math $\endgroup$ – user88576 Dec 7 '13 at 23:43
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There is little intrinsic value to studying mathematics chronologically, unless you are interested in the history of mathematics, as opposed to purely mathematics itself. You reasoned that learning the history of physics is interesting and can help 'flavor' the learning. Of course, the history of physics is not the same as that of mathematics -- I would consider the history of physics a little more interesting.

It is reasonable to study mathematics by moving gradually from the basics to the more advanced topics. It makes sense that this naturally carries you throughout the ages, as the basic ideas are older (at least in part) -- but there have been important revisions even to basic mathematics in recent history (for example, Zermelo-Fraenkel Set Theory). If you wish to simply build a good understanding of mathematics, then I suggest you disregard the history and simply focus on learning the math itself.

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    $\begingroup$ 'I would consider the history of physics a little more interesting.' - in your opinion, in your opinion $\endgroup$ – user88576 Dec 7 '13 at 23:30
  • $\begingroup$ @WissbegierigesWürstchen I thought about the statement, and I am prepared to stand by it. Physics has historically been much more politically controversial than mathematics. The history of mathematics is dominated by extremely interesting individuals, as is the history of physics. The difference, however, lies in that physics is a more applied field --- militaristic implementations of advances in the field, clashes with religious world-views, etc. are much more common in physics, and so it does not seem unreasonable to me to say that physics has had a more turbulent history. $\endgroup$ – Newb Dec 7 '13 at 23:52
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    $\begingroup$ Who cares about that it being controversial? This question is not really talking about the background, but about the actual mathematics itself. $\endgroup$ – Phaptitude Dec 7 '13 at 23:58
  • $\begingroup$ how come then that you ask this question? history is merely background, no math in it $\endgroup$ – user88576 Dec 8 '13 at 0:00
  • $\begingroup$ @WissbegierigesWürstchen I mean how certain stuff was done/proven/derived first, and for what reason (meaning was it part of a bigger problem or something similar, and if so which). What you're talking about is actual history itself. $\endgroup$ – Phaptitude Dec 8 '13 at 17:04
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I don't think so, here are the reasons why I think it's not:

First reason: mathematical methods for pedagogy have advanced through time. A lot of effort has been put to develop mathematical content which is useful for learning.

Second reason: efforts during the last centuries have developed new tools which are useful for analysing mathematical knowledge, simple examples include algebraic notation, the definition of a function, mathematical notion, and other things like category theory or abstract algebra which to my eyes are tools which are by themselves useful for learning new things. These tools give you a broader vision and "boost" you learning experience.

Third reason: Just because something was discovered a long time ago does not mean that it isn't going to be really complicated. For example: some Greek theorems in geometry are really complicated, while some basic results in graph theory are a lot easier to grasp (at least to my eyes).

EDIT: Of course, I think learning the history of mathematics is not only very important, but also very fun. I'm just saying learning mathematics in chronologically order is going to be very hard.(you're going to understand Ferrari’s formulas before knowing a graph has an even number of vertices with odd degree)

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  • $\begingroup$ disagree only about the "history of mathematics is not only very important, but also very fun", as the first statement is clearly wrong, and the last statement true only if you are >60 in age $\endgroup$ – user88576 Dec 8 '13 at 0:03
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    $\begingroup$ You mean the story of mathematics is not important and only people who are older than 60 find it important? Don't you find the life stories of Erdös,Galois or Ramanujan fascinating? I'm 18 and there are some things about the history i find fascinating. Its like fairy tails of the heroes who created all the theorems,conjectures and theory that we use every day. The stories of the giants on which we stand. $\endgroup$ – Jorge Fernández Hidalgo Dec 8 '13 at 0:13
  • $\begingroup$ Deciding which things are "fun" is surely a personal matter, to be decided by the individual who is (or is not) having the fun. $\endgroup$ – bubba Dec 8 '13 at 1:59
  • $\begingroup$ You are taking the question a bit too literally. I'm not asking whether everyone should travel back in time and obtain the exact manuscripts of Newton and Gauss, their arguments/proofs/derivations and motivation are the most important things. These 'new tools' are exactly what I'm trying to avoid, because in some cases they obfuscate the stuff, and if they don't you can use them in conjunction with the original literature. Your 'edit' confuses me, because now all of the sudden you say it is 'very important'. $\endgroup$ – Phaptitude Dec 9 '13 at 16:56
  • $\begingroup$ I think knowing the history behind mathematics is important because it helps us understand the background behind mathematical development, knowing it might help us recreate those conditions to get good results. but I do say in the post that its only my opinion. It is also my opinion that it is very interesting. I encourage people learn about the history of mathematics. $\endgroup$ – Jorge Fernández Hidalgo Dec 9 '13 at 19:15
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For studying math for the first time ? No ! For deepening one's understanding of math not usually taught in school ? Yes ! E.g., I've tried once, for instance, to find a parametric form for the sum of two cubes, only to find out later, (using Google, no less!), that Euler has already done it $300$ years ago! And the list of pointlessly wasted time and effort could go on and on and on, but I think suffice it is to say that quite a lot of it could've been spared, were I only to have known about and/or have had access to such vast and hidden treasures of mathematical riches ! Not to mention the fact that Archimedes more or less discovered calculus some $2,000$ years before Newton ! :-)

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Studying mathematical theories which were invented centuries ago is not necessarily a bad thing. In particular, Euclidean Geometry is still, in my opinion, the best way to be introduced to the notion of the mathematical proof, and of course, what is an axiom, a theorem, and why all these are important. Mathematical proof was born more that 2,500 years ago in the context of Euclidean Geometry, and thus it is not surprising that this subject still serves the purpose as the right teaching framework for introduction to mathematical proof. However, you do not need to learn ALL the old Mathematics. Calculus is inevitable, but it is presented in a quite different way from the way Leibniz and Newton presented it. Note that the modern and rigorous presentation of Calculus (i.e., with $\delta$ and $\varepsilon$) is pretty much due to Weierstrass (around 1860). Even Euclidean Geometry is presented today in a much more efficient way compared to Euclid's Elements.

Of all the important mathematical discoveries of the ancient greeks, the invention of the irrational is perhaps the most significant one. It was done about 100 years before the Euclid's Elements, which were written c. 300BC, in fact it is mentioned in Plato's dialogue Theaetetus, which suggests that it was known in c. 380BC, and that first proof was purely geometrical using "anthyphairesis", and not number theory. The number theoretical proof, which appears in the Elements, is pretty much the one we teach today in schools, but it is not the first one!

Apparently, I am not answering your question, but I do encourage you to read more, and in depth about ancient greek mathematics.

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  • $\begingroup$ It is interesting what is known to be the first proof in history ... $\endgroup$ – Widawensen Feb 1 '17 at 6:49
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In general, no, for all the reasons stated. You want to eat the omelet, not play with the hen while she runs around and doesn't lay eggs.

However as an enrichment, historical perspective is great:

*It is one more organizing thing to help ideas stick.

*Sometimes original papers are easier (not always, but see it in the hard sciences often).

*It is just of human interest.

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It is surprising how the greats followed historical approach. I will advocate both methods. Read modern textbooks when stuck search for the historical genesis of the idea. You may not need to go far back.

On a side, someone mentioned that the modern delta-epsilon definition of limit as better than Newton and Leibniz; logically yes, but pedagogically I disagree. Sometimes, in fact most of the times, the ancient reasoned like students at high school level hence pedagogically their proofs and explanations may be suitable for them.

The disadvantage of historical approach is however the learning curve. You might land in the land of the philosophers(esp. Aristotle's quibbles about the infinite, Rene Descartes new method of analytic geometry). To understand these(esp. Newton and Leibniz) you will need to be versed in the philosophy of the time(eg. absolute space vs relative space).

It is best to do these extra readings on your spare time. Read biographies more, find out which books inspired the great minds. And yes you will really get great inspiration that will be invaluable to understand abstract topics like group theory and topology. Trying to learn these without motivation is dreadful in my opinion. Professional mathematicians did not create these without a goal in mind.

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