Fundamentals of Advanced Mathematics

Lesson 0: Introduction


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Fundamentals of Advanced Mathematics
Lesson 0: Introduction

Understanding mathematics is a prerequisite to all other understanding.

This is because, unlike other forms of scientific understanding which use inductive reasoning, mathematics uses deductive reasoning.

Inductive reasoning: moves from specific premise to general conclusion.
Deductive reasoning: moves from general premise to a specific conclusion.

That is, generally natural and social scientists make observations of particular cases or phenomena and seek a general theory that describes or explains them. Mathematicians, on the other hand, use logic to draw conclusions based on the statements already proven as true.

Now, every mathematical principle is based off axioms.

Axioms: statements or propositions that are regarded as being established, accepted, or self-evidently true.

For example,
1 is the smallest number
two sets are equal only if they have the same elements
two parallel lines never intersect.

Now, for example, if you were to say that this last axiom was false, that would give you the non-eucledian geometry that you would study in a modern geometry course.

Mathematicians take axioms and the conclusions that necessarily follow from them to build mathematical theories. Thus the conclusions of a mathematician are proven to be true provided the assumptions are true. If the results of a mathematical theory are deemed incompatible with some portion of reality, the fault is not in the theory but with the assumptions about reality that make the theory inapplicable to that portion of reality.


Let’s take as an axiom that all men are mortal. This is obvious, no one has ever heard of or seen anyone immortal, but we cannot prove it logically.

Let’s also take as an axiom that Socrates is a man. Again, this is certainly obvious but is in no way logically prove-able.

However, given that all men are mortal and that Socrates is indeed himself a man then we can make a deductive argument and say that Socrates must also be mortal. That is, we have proven that he is mortal based on our two axioms. If it is later shown that Socrates is some sort of immortal entity, then the error is not in our argument that he is a mortal, but in our axioms. Either all men are not mortal, or Socrates is not a man.

This course will teach you to think like a mathematician; to think deductively. There is no prerequisite. Three types of people will take this course:
One: people with absolutely no mathematics background past the mandatory high school algebra. Though this may give you problems in other high level mathematics courses, that is not a problem here. Indeed, this may even be a benefit; you will be able to study the language of advanced maths without being tainted by computational drudgery and formula memorization.

Two: people who have taken elementary college level courses, typically including calculus differential equations and linear algebra, and are interested in expanding their mathematical knowledge. This course may feel a little different than any other math course you have taken. Up to this point, your courses were probably based on systematically solving problems, and thus emphasis was placed on the computational tools and algorithms for this effect. In more advanced mathematics, however, the emphasis is placed on concepts and their connections; this requires intense thought, intuition, and understanding of the objects at hand. Thus you may have to forget everything you thought you knew about math and re-compile your understanding. This is a benefit, because now you can own the understanding.

People already acquainted with advanced maths. This course will be an easy overview for them and they will be able to contribute content and commentary for everyone else.

When it comes down to it, you simply cannot learn advanced maths by watching a video. If that is your goal, thanks for watching, quit now. Learning advanced maths requires a determination to really understand the topics covered. It requires accepting nothing anyone tells you as truth until you have convinced yourself of it’s validity. That is you have to own the understanding. Make it yours. You should be able to say ‘I understand the difference between for all there exists and there exists for all’ not because you memorized the difference but because you sorted through the logical difference on your own. For this reason I strongly encourage you to spend more time in the problems section then the video section. These are not the type of systematic problems you’re used to. In general you will be asked to prove something based on something else. The most important aspect here is the act of drawing the logical conclusions on your own. Repeating this process is what will reshape the way you think and the way you interpret the world around you. This is the lasting value of studying mathematics, everything else is insignificant in comparison.

After you have thoroughly digested a problem, share you solution with others; talk about how you arrived at it and help others who are struggling. Remember though, you are wasting your time if you simply look up the answer before thoroughly considering the problem.

I am not here to teach you in the sense of presenting knowledge for the purpose of memorization. I am merely a guide. I will present you the language and topics but it is your responsibility to own the understanding–to sort through the logic to understand fundamentally what is going on.

Thanks for watching, This is Math.