One Approach to Understand Concepts Names in Mathematics

Yes, you can!

One of the major hurdles to understand math concepts appears to be, as I call it, the 'misleading language' often used by mathematicians' community.

How many times you stared at some mathematical term or equation, and had a feeling you are in front of a wall you are required to climb and go over it, or, even worse, go through it, but you don't have any tools at your disposal to do so? And, perhaps, on top of that, you are pressed by time, for instance during an exam. You felt helpless, with complete absence of any groundbreaking idea how to proceed forward.

And, how many times you questioned your mathematical ability by considering your school test results? I have news for you - they are not the only measure of your mathematical powers!

So, you are frustrated; you are helpless; even scared what will come if you don't know the math!

Well, fear not anymore! I have a remedy for that situation! I have a cannon for that wall! After my explanation of the core challenge, and when you combine that knowledge with my Postulate driven Implicative Truth Synthesis method, you will be in a position to significantly advance in mathematics. You will be able to tackle almost any mathematical text and, after some research, fully understand what is it about. Soon you should be able to manipulate numbers, matrices, mathematical curves like real tangible objects in your mind, rotating them in 3D, thus visualizing quantitative relations for better processing.

My central solution idea may sound simple and obvious but many great idea often looks like that! The core of the issue is the habit of mathematicians to label, name mathematical concepts using words from our real human experience, from everyday life, or using the names of people. With this approach, mathematicians not only mask the core of the meaning of the math concept, but also misguide the reader to a different and ineffective train of thoughts. You really don't know what is worse – naming a math concept by some famous mathematician, or use ordinary word from everyday life with misguided association that leads into indecisiveness and confusion..

So, let's start with shading light on this matter!

To form a correct relation between a name or label for mathematical concept we should first clarify what a mathematical concept is and what it can be. By answering this we will build a foundation idea of everything else that will come later as well as how to deal with an arbitrary name given to the mathematical concept.

Mathematical concept is almost exclusively a set (or sets) and the sequence of set operations on them. Every mathematical concept can be explained and represented by sets and related sets operations. We all know sets, right? They are simple concepts. And operations are straightforward. You add elements to a set, or remove elements from a set. You can combine sets, find union, intersection, etc.. Now, to develop mathematics into different branches what do we need? Sets, of course, and specific, postulated sequence of operations on them. We are free to postulate sets. We are free to postulate operations on them. By doing this we can, actually, define starting points of any branch of mathematics. And going in the other direction - when we see a mathematical symbol and formulae we immediately should think what are the sets that are postulated, and what are the postulated (given) operations on them. However, often, we will not need to go all the way to the sets, although it won't hurt. Instead, we can think of numbers. Then, again, every number is a collection of something, a count, a kind of set – hence, we will be dealing with sets of numbers.

Here are just some of the mathematical concepts that can be fully described by sets or sets of numbers and operations on them:

function – pairs of values; this definition is not concerned how the pairs are obtained. Another definition is a map or association of two or more numbers. Functions are generally postulated. You often do not need a proof why you chose such and such function. There are infinitely many functions and hence infinitely many postulates. Even when a function describes behaviour of a physical object or process it is still a postulate, starting point as mathematics is concerned.

Here are some examples:

integral – way to sum up certain numbers you specified or postulated. Numbers are counts and always come from sets. The operation of integration is also postulated. So, in terms of sets or numbers you redefine integral as a specific way to sum up numbers.

equation - compares sizes of sets.

differential equation – equation that uses differences of some function

matrix – selected or calculated numbers displayed on the page in an rectangle. The rules are specified what to do with these numbers. You can do whatever you want but you often chose to do something you can benefit from.

norm – a way to define a size of e set which can be number, function, matrix.

metric – distance between mathematical entities (functions for instance) similar to the dsitance between numbers or difference of set sizes.

convergence – approaching certain set size by calculation or by manually removing or adding elements to the set. You need norm and metric to achieve convergence. Approaching certain number by picking closer and closer numbers or obtaining closer and closer numbers by some calculation (operation).

derivative – certain, postulated sequence of operations on functions, which in turn are operations on set of numbers, which in turn are operations on sets.

compactness – misleading nothing-to-tell term for some property of sets of numbers and sequences of numbers.

limit – picking elements of sets or numbers arbitrary or following some rule.

space – set of numbers, functions, vectors, or matrices with certain numerical properties.

As an exercise, find a good mathematical dictionary and for chosen terms try to give your own definition using only sets, numbers, and operations on them. Ask yourself, what numbers or sets are postulated in this term and what operations are specified for them.

With our new method we can formulate alternative way of looking at the mathematical formula. No matter how the symbol may look strange, intimidating, or a name that tells nothing about the underlying concept, we should only look to which postulated sets and postulated sequence of operations on them this symbol refers to. We will completely ignore the name and, instead, look to what kind of sets and sequence of operations on them it refers to! Once we know that, the name is not important (never was). Name has a convenience that many people may know or learn about it. But, that's the only positive thing about it. Name is only a shorthand. You can fully explain any mathematical concept without using any name – only by only using sets and operations on them!

To tackle apparent abstraction and complexity of mathematics it is prudent to consider the following: everything in mathematics is about sets and operations on them. Any mathematical formula, branch of mathematics, or mathematical operation can be traced and explained in terms of sets. The trouble to understand mathematics is in the fact that mathematicians go to a great length in naming sets and the sequences of operations on them. Calculus, probability, measure theory, random numbers are just labeling names for sets and the sequences of operations on them. Good news is, you don't need to use names given by mathematicians to advance your knowledge in mathematics. Instead, think in terms of sets and use any labels that work for you, knowing that in the background it is all about the sets and operations on them. And moreover, you don't even need to use language to think about mathematics – think in terms of sets, and use language only to try to convey your results to others.
   

We often hear that some results are obtained by using mathematical logic. But the concept ”mathematical logic” is misleading. It suggests that there are many kinds of logic, one of which is mathematical. However, it is not true. There is only one logic for which we can say it is applied to different disciplines. Logic is not more precise if it is used in mathematics than when it is used in a courtroom or during scientific research in chemistry. Logic is about examining truth values of the statements about world around us. Logic deals with truth values and manipulates truth values in any discipline or discourse of thinking.

     

Any discipline, being it mathematics, physics, jurisprudence, chemistry, architecture deals with truths within that discipline. These truths are related to concepts and characteristic of the particular discipline. We use universal logic to discover truths within any discipline – for mathematics we use logic to discover truths about numbers. One of the reasons mathematics appears to be more complex and alienated from our ways of thinking is because mathematicians arbitrary name their steps in discovering the truth and the truth statements. The names can be quite unexpected and exotic, and they can foggy the clear picture why they are there – to define sets and operations on sets. have yet they refer only to numbers, truth about numbers, and operations on numbers. Mathematics appear to be more precise than other sciences. Hence, in mathematics we have theorems, lemmas, proofs, definitions. But, they all relate to the efforts to discover truths about numbers. I am sure there are theorems and lemmas in chemistry and architecture, but they may not be called the same name.

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