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We are talking about the subject Newton created to be able to even talk about physics here! Reading the whole paragraph as reported above, it is clear that it is quite different from the title of this question. Saying that "X-theory involves no application in engineering" just means that an X-theorist, in her job, doesn't employ engineer's tools or language, and as an X-theorist she may even forget about engineering. It definitely does not mean that i problems of X-theory were not originated from concrete problems of engineering, nor that ii the results of X-theory have no applications in engineering.

Actually, at the origin of even the most abstract mathematic theory there are concrete problems of applied science maybe after several successive steps of abstractions , and also, the final applications are again back in concrete problems. Abstraction from abstractus : p. While I agree that the paragraph is largely a sales pitch, I think it does hit on something else. It says that real analysis doesn't involve applications to other science.

I take this to mean that when you are doing or studying in a first course real analysis you don't look at applications to science. This is in contrast to calculas, whereas many of the problems in calculus books are focused on all kinds of problems from classical mechanics and other areas. Just a final note. I thing that Pugh's book is amazing, the best undergrad analysis text out there.

There’s more to mathematics than rigour and proofs | What's new

Mainly because of the HUGE number of very good problems. Davidson and Allan P. The book is divided into two parts. Part A deals with "Abstract Analysis" which includes theory, proofs, examples, and problems found in most undergraduate analysis books. Part B deals with "Applications" which include polynomial approximations, discrete dynamical systems, diff.

The comment on the back of the book seems to be saying "This is a math book. To the extent that the book is a pure math, it is not a book about physics, horticulture, or sociology. But so what? You could say the same about any area of math. This seems like a particularly odd thing to say about analysis since it can be applied so directly to other areas.

Secondly,despite loving Pugh's book-I call it "Rudin Done Right"-I was also very disappointed at the very terse preface. You'd think someone with Pugh's teaching experience would have a LOT to say on the subject having taught so many years to some of the best students in the world.

Third-I seriously doubt one of the world's experts in differential equations thinks real analysis is devoid of real world applications. But that being said-what compels people teaching this course to strip it down to Bourbakian purity? Or something darker and deeper? I'm waiting for a balanced text at this level that unifies physical applications with a comprehensive introduction to real analysis.

If it never arrives,I may have to write it myself. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Real analysis has no applications? Ask Question. Asked 9 years, 3 months ago. Active 2 years, 7 months ago.

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Viewed 28k times. Did you like proving theorems? Are you sick of memorizing integrals? If so, real analysis could be your cup of tea. William Lawvere said that a good idea may be reinvented many times. So there are many words for these arrows, and you may have come across the idea of an vector to represent this idea. So our states are two, the beginning set, sometimes called a domain, the end set, sometimes called a codomain, and their transformation, an arrow.

A First Course in Real Analysis

Arrows are very important in mathematics. We can have an arrow at the top, an arrow at the bottom, and an arrow representing the transformation from the arrow on the top to the arrow on the bottom. This is called a functor, but usually we link the base and tips of the arrows each with an arrow, and this is bad.

What we need is one arrow at the top, one arrow at the bottom and one arrow not two transforming the top arrow to the bottom. I tried to explain this to a three year old in a train. We came across birdie, tunnel, horses, cows, arrow and functor. I am not sure one way or the other whether I succeeded, but it did not end in tears. Maybe I needed more examples.

The other thing we might have is numbers and transformations of numbers. There are many examples of this. Sometimes the transformations are called functions. So we start with a set of numbers, represented by a line, and we have the number zero on it, and then if zero goes to zero, we have another line at right angles to it with the zeros at the corner, and the function is just a wavy line that matches the numbers we began with represented horizontally with the numbers vertically.

In fact for a wavy line two different numbers represented horizontally may have the same number vertically. If we go backwards from vertical to horizontal numbers in this case, it is not called a function but a multifunction. This is because one object represented vertically can transform to two numbers represented horizontally. The interesting thing is when this is multifunction in both directions.

Say we had a connected piece of string on a plane and it was all over the place, with the string intersecting itself. Then this is a multifunction from horizontal to vertical, and a multifunction from vertical to horizontal. In fact, if we started from just the horizontal to vertical multifunction, and kept bouncing the transformation in both directions, then if we continued until we had no more numbers generated I call this a bouncing set — it might be infinite then we would have a simple function from two bouncing sets, say from a horizontal bouncing set to a vertical bouncing set.

These bouncing sets are examples of multiobjects. Having given these examples, and supposing you can frame what you want to explain in these terms, and it may be difficult, or it may need training from your teacher, or it may be impossible because the ideas cannot be framed in these terms, then you want to explain what you perhaps know. I think there are two stages. First to describe to yourself what you know. If you cannot describe to yourself what you know, you are unlikely to be able to describe it to others.


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So lets leap a stage here and assume you know what you are talking about. Here we hit a problem. Even some really famous mathematicians know what they are talking about, but cannot explain it to others. A really interesting case is the famous mathematician Riemann.

In his work on number theory he came up with an interesting statement which he thought other mathematicians could easily work out, and I think it was about 50 years before, after much arduous calculation, anyone else was able to prove it. A friend of mine, Doly Garcia, describes me as having transparent bran syndrome. This is the same thing.

It is a mathematical disease in which the owner or developer of a mathematical idea assumes that the listener has automatic access to his or her brain. Unfortunately, telepathy is not at present with us, and one of the prime tasks of a mathematician is to go through all the boring detail, including really minute detail, and explain everything as if it were to a three year old. We have already seen that three year olds may in the future understand functors. Indeed I think this is essential to the continuing development of mathematics.

There are people called serial killers. When writing an explanation, be a serial amender. There are many ways to improve what you write. Reduce the unnecessary bits. I am not explaining this, but I call this syntactic fluff.

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It is similar to what happens sometimes when you are asked to write an essay within a set number of words, and you go over. You have to go through and delete what is necessary to fall within the limit.


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The opposite process is to go through and look at a technical idea which maybe you have mentioned in passing, and decide to include at least an indication of what you re talking about. This inclusion of more detailed meaning expand the text. Finally, if you have done all this, and you go through and you realise what you have written is difficult to understand, you have failed. A god idea is to go through each word and see if you can simplify it, particularly jargon words. There is far, far too much jargon in mathematics.

The object should be to explain ideas so that knowledge can increase and people understand.