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If it is the case that I wish to concentrate my efforts in learning Probability, Statistics and Game Theory, then is it also the case that I must become proficient in calculus?

Is calculus a foundation for all mathematical learning... or are there other foundations that would better pertain to Probability, Statistics or Game Theory.

I ask this because

-When reading books such as "Introduction to Probability" "Introduction to Statistics" or "Introduction to Game Theory" I always end up encountering terminology or notation that I cannot understand and therefore cannot progress through the rest of the book. I am unfamiliar with the terminology/notation and my immediate reaction is that these terminology/notation are related to calculus in some way ---> am I mistaken in this assumption? Are there mathematical foundations outside of calculus?

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    Probability is a very broad subject. Can you be more precise? Can you, for example, list some of the terminology and notation that you don't understand? Have you tried looking them up on Wikipedia?2011-07-19
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    I have looked most of them up on Wikipedia. The biggest problems that I have been facing were associated with set theory. Almost every time set theory came up infinite sets followed and the summations started to pop up (as if it was supposed to be assumed that I understood their meaning) this is what caused me to stop reading literature on probability or game theory and focus just on calculus. Which brings me to the question of whether this was the right choice. I'm unfamiliar with how to represent symbols on this sight so it's hard to give examples now.2011-07-19
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    Even the use of subscripts and superscripts began to confuse me. I've searched for an introduction to set theory, but it seems to cause more problems because there's no basic explanation for the use of any of the notation... this might have to do with my lack of experience with higher level math, but it's hard to find a bridge. I've tried to google and wikipedia the notation for the topics, but each explanation for these notation or terminology seems to require more assumed knowledge.2011-07-19
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    Well, it is quite important to understand infinite series in probability, and generally the first place you learn about infinite series is in a calculus course. Are you a high school student? Is there a school nearby where you could take higher-level courses?2011-07-19
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    I've just graduated high school and am going into college. The college I'm going to be attending will of course have higher-level courses. Your suggestion would be for me to take calculus then?2011-07-19
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    Yes, that would be a good idea, and there should be probability and statistics courses to take as well after that.2011-07-19
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    Thank you very much for the help. I really needed the confidence to commit to learning calculus and I have definitely obtained that.2011-07-19

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You're asking several related questions here, and some of them I'm not sure you're asking, but here's one I'm pretty sure you're asking: no, calculus is not the only foundational thing you need to learn in mathematics. Even very applied mathematicians should learn lots of linear algebra as it is quite ubiquitous, and it is always a good idea to know a little analysis (roughly, rigorous calculus) and topology. Depending on what you're interested in, it's also a good idea to learn some abstract algebra, maybe a little number theory, and some more analysis and topology. And depending on what kind of probability you're interested in, it would be a good idea to learn some combinatorics as well.

This is not a complete list, exactly, but it's a start. There's a lot of mathematics out there.

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Well, calculus is not a tool for intuition, it's name says it : it's meant to calculate. I believe the purpose for calculus is that when our intuition fails to explain things, we rely on theorems and then use calculus to confirm our intuition. We don't use calculus as a fundamental : it's a tool. The fundamentals lie behind the ideas of those theories. For instance, probability without calculus is just a bunch of nice ideas, because there's always series in there, integrals, and such, so you need to calculate stuff if you want to be able to say something interesting...

EDIT : As an example, so that downvote can be removed...

Guessing the expectation of a continuous random variable can be a pain if the variable has a non-trivial expression, and probabilistic theorems might not make it easier to solve it : using calculus to compute the integral might make it work. Here, calculus helped where probabilistic intuition couldn't ; that doesn't mean there is no intuition in calculus. That is what I meant by "Calculus is not a tool to develop intuition. It's a tool for computing."

All this was said in a probabilistic context. Calculus lovers, I am not offending you : calculus is amazing on its own, too.

Hope that helps,

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    That was awesome. That did help a lot. I'm more excited and motivated to learn calculus now that my mind has settled on it's necessity.2011-07-19
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    "We don't use calculus as a fundamental : it's a tool. The fundamentals lie behind the ideas of those theories." This makes sense now.2011-07-19
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    @Derek: you should show your appreciation of an answer you like by upvoting and, if you like it enough, accepting the answer. It makes us feel good.2011-07-19
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    Haha. Gotcha. Will do. I need more reputation... will definitely remember that though.2011-07-19
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    A great deal of intuition accompanies Calculus. That is it least as powerful as its ability to compute exact areas, volumes and other quantities. It is an acquired intuition that is extremely valuable. I find myself 100% at issue with this statement.2011-07-19
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    lol... I guess we'll have to wait for a response from Silva. I would question both of your uses of 'intuition'. Maybe Silva is assuming that intuition is not something that can be acquired... intuition and calculus being separate tools that have uses that are mutually exclusive. Thank you all very much for your help. I did not expect the amount, quality or quickness of the responses... good stuff.2011-07-19
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    ncmathsadist. Your comment about acquired intuition is interesting. Could you elaborate? Do you mean to interpret intuition as an ability that uses quickness as a measurement for answering problems. My initial interpretation of intuition had to do with some sort of a priori knowledge that allowed answers to questions without experience. I like the idea that I can sharpen my intuition through practice.2011-07-19
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    Uhhhh okay I disagree with your downvote. @ncmathsadist : I never said there was no intuition WITHIN calculus. I said that you're not using your calculus intuition to deal with other areas of mathematics ; when you use calculus in probabilities or game theory, you just compute, that's it. It doesn't mean that the theorems and methods that have been developped in calculus are intuition-less, in fact, they have quite powerful intuitive ideas, just mentioning Stokes' theorem as an example.2011-07-19
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    @Derek : To remove confusion, I consider intuition as getting a feeling of how all the theorems you have proved are proven, and why they work : having this idea of the mechanics behind your arguments lets you pull off some new ideas whcih can make your problem solving easier. Usually, computing something using calculus in other areas of mathematics (as probability, game theory, etc) helps you get out of trouble when intuition can't give you a good idea to solve your problem.2011-07-19
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    For instance, if your probabilistic intuition can't give you an idea to compute an expectation of a discrete variable, using the definition of expectation and computing a serie or an integral can give you an answer. That is what I meant by "Calculus is not a tool for intuition. It helps you when you don't have intuition." By the way, thanks ,mixedmath.2011-07-19
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    @Patrick: No problem!2011-07-19
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    @Patrick: Thank you for the clarification. I understand your use of intuition better now.2011-07-19