This is very general question I am asking here. But I think it really needs to be addressed. Whenever I come across a new concept in mathematics, I try to understand it by searching on the internet. It is always the case for me that when I do find some stuff about those concepts, many times they are elaborated heavily with complex notations and highly formal language.I think a person with some/no mathematical background can never get it. Thankfully there are blogs/sites which explain it in more natural language help me not only understand those concepts but also show the beauty behind those mathematical concepts. Don't you think that extra formality and over usage of symbols really drives away people to go deep into those concepts and hinders the progress of mathematics as less people get involved. No branch of science has progressed with just few peoples getting involved. It is always seen that as more and more people get involved, more progress is done. We have seen that open source softwares evolve way better than proprietary softwares.
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6Try proving something without formally defining it, and you'll see why. – Alex Becker Feb 24 '13 at 22:04
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1There's a limit to how much you can do without formality. One rather famous example of the downfall of informal mathematics is Naive Set Theory. From my experience, mathematical rigor is normally only forced upon actual mathematics majors anyways. The mathematics taught to the general population is still rather informal. In pure mathematics it is needed both to define concepts without ambiguity and to enforce careful thinking. – EuYu Feb 24 '13 at 22:05
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2It does not have to be too formal. It has to be just formal enough. – Trevor Wilson Feb 24 '13 at 22:07
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- Whenever I stumble upon a new Swahili word, I google it and am unhappy because it usually appears in pages full of Swahili words I don't understand. - 2) Mathematics is not proprietary as you see from finding explanations freely available online or in libraries. In fact, math lives from open collaboration. When was the last time you read through the linux kernel sources line by line? (Math is very userfriendly, it is just picky whom it considers a user) - 3) "We" mathematicians try to get as many people involved as possible, but must shrug off before mastering finitedimensional calculus
– Hagen von Eitzen Feb 24 '13 at 22:10 -
1You seem to think that mathematics is difficult because we use lots of symbols. That's not true. Mathematics is difficult because the ideas are often complex and deep. Symbols are there to make things easier and more precise. – wj32 Feb 24 '13 at 22:11
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1I think this is relevant: Why do mathematicians use single-letter variables? – Feb 24 '13 at 22:12
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"I think a person with some/no mathematical background can never get it." Yes, and this isn't a bad thing. We should expect that mastery (or at least competency) of the basics is required before moving on to more advanced topics. I remember reading an article about the Clay Millennium Problems back in 2000 when I was in fourth grade. I remember sitting down and trying to solve the first one at my kitchen table. Boy was I overreaching... – Ben West Feb 24 '13 at 22:13
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@BenW. so you soon switched for the second problem? – Hagen von Eitzen Feb 24 '13 at 22:17
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1Math has never progressed by getting everyone involved. It progresses by getting a few, extremely smart, extremely well trained people involved. Most modern math problems are so complicated that without spending years of training building up to it, a regular person wouldn't even be able to think about the problem in a useful way, much less actually solve them. And the majority of people simply aren't interested enough to go through all the training required. – AJMansfield Feb 24 '13 at 22:18
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2It's truly a shame that questions like this are so quickly closed ($11$ minutes after asked). One can certainly give good arguments to support the value of formalization in mathematics. These are questions that commonly occur to educated laypersons or mathematics students interested in foundations and philosophy. We should strive to give them thoughtful answers. The question needs a few reopen votes. Please contribute. – Math Gems Feb 24 '13 at 22:19
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@HagenvonEitzen Ha! I decided it'd be better to go back to my long division homework problems. – Ben West Feb 24 '13 at 22:21
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2Why was this question closed? How is asking for the reasons behind the formality of mathematical rigor contrary to the intentions of this site? – CogitoErgoCogitoSum Feb 24 '13 at 22:22
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4@MathGems: As written, the question is basically a rant. In fact, the FAQ specifically says not to ask questions of the form "I’m curious if other people feel like I do". If the post is rewritten in a calmer tone so that it asks a proper question, I would be happy to vote to reopen. – Zev Chonoles Feb 24 '13 at 22:23
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1@AJMansfield This reminds me of a quote by Eugene Wigner on modern physics: "Physics is becoming so unbelievably complex that it is taking longer and longer to train a physicist. It is taking so long, in fact, to train a physicist to the place where he understands the nature of physical problems that he is already too old to solve them." – EuYu Feb 24 '13 at 22:24
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1@Zev Why did you unilaterally close it before it reached $5$ votes. This should be a community decision. You thought it was worthwhile enough to answer, but still closed cast a binding close vote??? – Math Gems Feb 24 '13 at 22:28
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1@Math Gems: I had no idea that "uni" now meant 4. Sorry everyone, the question should have been open an extra few minutes until it collected a fourth vote to close. Then it would have been a community decision. I guess I'm just a tyrannical dictator. – Zev Chonoles Feb 24 '13 at 22:30
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1@ZevChonoles While I agree with your decision to close, your last comment is a bit mean. – Alex Becker Feb 24 '13 at 22:31
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@Alex: Take a look through this thread and note that Math Gems has accused me and the other moderators of that and worse. – Zev Chonoles Feb 24 '13 at 22:33
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@MathGems: Yes, that is exactly what I thought and did. – Zev Chonoles Feb 24 '13 at 22:34
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2@ZevChonoles Ah, I temporarily forgot who "Math Gems" actually was. I withdraw my complaint. – Alex Becker Feb 24 '13 at 22:35
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2@Zev As a matter of principle, binding close votes should not be cast by moderators who have just answered the question, because it could possibly be viewed as a way to game the system (preventing other answers after one has answered). One could argue the same for anyone who has answered the question. But it is especially important for moderators to set good examples. – Math Gems Feb 24 '13 at 22:40
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4@ZevChonoles I don't care about "gaming the system" or whatever, but it does seem to me that one should either think a question worth answering or not, and only answer it in the former case and only close it in the latter. – Ben Millwood Feb 25 '13 at 00:34
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@Ben: Almost always, I would only do one or the other. But I don't think it's contradictory to post and vote to close. It can be preferable to responding to the question in the comments (as people often do for off-topic questions, and as several people did do above) because it separates the discussion of the responses to the question from other discussion (such as the one we're having right now). – Zev Chonoles Feb 25 '13 at 07:44
2 Answers
It's true that we can talk about math informally in order to gain intuition and understanding. We can discuss examples. We can explain how we think about the math.
But when it comes down to it, everything still has to be stated precisely, and for this, formal language is necessary. That's the beauty of mathematics - everything is rigorously defined, to infinite precision. The exposition by itself is not enough. How can you prove that two lines intersect only at exactly one point if we don't define what a line is? We can say intuitively that it should be that way - but we can't prove it until we say what exactly a line is and what exactly it means for two of them to intersect.
You are probably frustrated because the internet is not, in general, a very good place for motivated mathematical discussion, especially in lower level disciplines. When people google about math, they're generally searching for homework solutions, so the solutions, without much explanation, tend to be what are most commonly available.
Books are much, much better for this. Many people who write math books try to motivate the subject and provide helpful exposition on how to understand the complicated ideas they present.
It takes more patience to read a math book than a short article, but that's just the name of the game. Math is hard, and it's silly to think that there should be some presentation which allows anyone to immediately understand it without careful thought. For that reason, math will probably never be as popular as watching TV or listening to music. But, that's just how it has to be. I still like it.
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4+1 for your mention of homework solutions. It is there where an unnecessary abundance of formal language is found (quantifier bloating, rightarrows and therefore symbols for inference, ...), even when good mathematical writing would add a lot of prose (though keeping in the back of ones head that an undisputable formal translation is available) – Hagen von Eitzen Feb 24 '13 at 22:15
To steal a good alliteration from Gauss: the notions of mathematics require intuition; the notations of mathematics require precision. If someone isn't able to understand (much less formulate their own) precise logical statements and arguments, either in symbols or in "jargon", they aren't going to be contributing anything to mathematics research. There's no reason that just getting "more people" involved will help mathematics (or any other field for that matter).
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