Monday, February 27, 2012

Interrelations Between Deductive Systems and Inventive, Innovative Thinking

Looking at one deductive system, call it A, in the context of other deductive systems, can show how these extraneous systems motivates development of the system A. By being "in context" I mean that theorems in one system, which has presence in the logical, conceptual surrounding of the system A, hence providing context for it, are axioms or starting propositions for theorems in the system A. The systems need not to be mathematical only. What is important is that the systems are based on deductive reasoning, and that they are axiomatized as much as  possible. I allow inductive reasoning, and definitely intuition as a method of discovery, but eventually, these both approaches will be morphed into a deductive structure and method. I wouldn't even differentiate inductive reasoning from deductive, but rather call it "dynamic deduction" or "deduction with self error correction".

This kind of deductive systems linking, where contextual nesting and inclusion can go infinitely (i.e. any system that provides context for system A can itself has its own context, etc), is a core of inventive, innovative thinking.

Saturday, February 25, 2012

About Number Definition, Pure, and Applied Mathematics

If we agree that math is about counts, and counts only (as it is, since numbers, counts come from the cardinality of sets and set theory) then geometry doesn’t belong to mathematics. It is, by some authors (mentioned in "What is Mathematics Really", R. Hersh), considered impolite to have any geometric drawing in a mathematical text. Geometry has link to mathematics as the morning purchase of vegetables on the local market has. The geometry is only more convenient (perhaps!) in representing numbers and their relationships. Geometrical figures do only one thing to mathematics – by measuring the distances, angles, etc. we generate numbers, and sets of numbers. None of ZFC axioms refer to anything geometrical in the same way that ZFC axioms do not refer to the bunch of carrots at the local produce markets.

Geometry can help to visualize certain mathematical relationships and results. But, the link between pure numbers and sets to the geometry is in essence arbitrary. Geometric interpretation of mathematical results are neither mandatory nor necessary.

For mathematics, it is completely arbitrary what or who generates numbers. The process of numbers selection, generation, numerical operations can be scientific, guessing, or a product of any dogmatic philosophy. Math couldn’t care less. As for geometry, the reason it has a strong presence in mathematics is just because of some of its practical applications. The reason why we can abstract real world into points, lines, planes, spheres is extraneous to mathematics. For whatever reason a line is drawn, and for that matter, what that line represents abstraction of, is not a part of mathematics. From math point of view we draw lines to generate numbers by measuring the lines’ lengths. Measuring process (with instruments, visually, or in any other way) again, is not part of mathematics. Math will see only the number you obtained.

For example, when we write 2 x 3 = 6 (without any explanation) will the reader know where 2 and 3 came from? Of course not. It can be from 2 baskets, each one having 3 apples. Or, it can be from 2 cars, where each car has 3 passengers. Why do you need a rectangle with sides 2 and 3 to explain you this mathematical result? You don’t need it.

As much as apples, cars, are not part of mathematics, in the same way is not rectangle or any other geometrical figure. Geometry is perhaps interesting because it selects, generates certain sets of numbers that are of interest in everyday applications, like lines, squares, rectangles, circles. It is quantification of these figures and their measures that matter to mathematics, and not figures themselves. The thought  process that takes place in defining a circle as an ideal abstraction of all real world attempts to make a circle (as well as a straight line abstraction of all straight directions) is a nice thing to think about, but that’s not part of mathematics. Once you “idealize” circle, math cares only about the numbers you provide by measuring them.

Simply put, no geometric figure should be considered an element or part of pure mathematics because none of the theorems in math are proven using them. If seemingly geometry terms are used in proofs or appear to be a focus of study, like trigonometry or differential geometry, it is because the axioms of geometry are part of it, but, they are not part of mathematics. Mixing ZFC axioms and geometry axioms is like mixing ZFC axioms and axioms of any other system, including "marbles used in counting", carrots methods of purchase, quantitative finance rules, etc..
"The formalist makes a distinction between geometry as a deductive structure
and geometry as a descriptive science. Only the first is mathematical. The use of
pictures or diagrams or mental imagery is nonmathematical. In principle, they
are unnecessary. He may even regard them as inappropriate in a mathematics
text or a mathematics class."  ("What is Mathematics Really" Rueben Hersh)

Of course, it doesn't mean you should not use them to better communicate your ideas, investigate new directions in math or other sciences, or visualize a bit more difficult concepts in mathematics.But, you have to clearly differentiate between mathematics and these non mathematical objects and concepts.

Mathematics is defined not by objects it counts, nor by reasons or logic why those objects are counted, but with concepts used to define mathematics axioms and to define proofs of mathematical theorems.
[ to be continued...]

Monday, February 20, 2012

From Reuben Hersh's book "What is Mathematics Really"

From Reuben Hersh's book "What is Mathematics Really".

Any proof has a starting point. So a mathematician must start with some
undefined terms, and some unproved statements. These are "assumptions" or
"axioms." In geometry we have undefined terms "point" and "line" and the
axiom "Through any two distinct points passes exactly one straight line." The
formalist points out that the logical import of this statement doesn't depend on
the mental picture we associate with it. Nothing keeps us from using other
words—"Any two distinct bleeps ook exactly one bloop." If we give interpretations
to the terms bleep, ook, and bloop, or the terms point, pass, and line, the
axioms may become true or false. To pure mathematics, any such interpretation
is irrelevant. It's concerned only with logical deductions from them.
Results deduced in this way are called theorems. You can't say a theorem is
true, any more than you can say an axiom is true. As a statement in pure mathematics,
it's neither true nor false, since it talks about undefined terms. All mathematics
can say is whether the theorem follows logically from the axioms.
Mathematical theorems have no content; they're not about anything. On the
other hand, they're absolutely free of doubt or error, because a rigorous proof
has no gaps or loopholes.

Sunday, February 19, 2012

The Definition of Number

After a number of years dealing with mathematics in your primary, secondary school, there still may be a question what the number is. Moreover, unless you are a professional mathematician, with PhD in your resume, I can safely assume that your frustration and fear of mathematics is still present.

What is a number? Seemingly popular approach I am using here does not reduce the strength and significant clarity of the definition. Bear with me, and listen carefully :-) You may find out many interesting things!

Here is the clearest approach to defining number.   

Number is a count.

I will repeat again, number is a count. The purity and significance of this definition can not be emphasized more. While it is simple, it conveys many more important messages than other definitions and approaches you may have read about before. One of the most important message, in my view, of this definition is that it implicitly specifies what you can do with counts. Knowing what you can do with counts, you actually filter out all non mathematical concepts that may be mixed during "bad" mathematical lectures over the years. Also, thinking of numbers as counts, you define what pure mathematics is about! And that can help you answering the questions how math can be applied (about what "applied" means we will see later) in so many different fields, and what differentiate pure and applied math.

What you can do with counts is what mathematics is all about! So, what can you do with counts? You can add them, subtract them, divide, multiply. You can, then, do any number of these operations in any sequence you want. No apples, pears needed to do that! Count 5 is a universal count. It can come from counting apples, pears, cars, atoms, money, steps, seconds. That number 5, count 5 is a universal thing for all of them. While you can eat 5 apples, drive 5 cars, wait 5 seconds, with count 5 you can not do that. But you can add another number 5 to it! or deduct count of 3 from it. Or do any other "counting" operation! Note very important thing -> how you call these counts, i.e. are they integers, positive, negative, odd, even, rational, are just labels we attach and associate to the concept of a count! There is only count we are dealing it all the time.

Looking at count 5 only, i.e. number 5 only, you can not tell whether it came from apples, pears, or counting seconds. So, how you will differentiate count 5 of apples and count 5 of seconds if count, number is actually so universal concept? There is no other way than to keep track by yourself what you have counted. Technically, you will write a small letter beside the number, beside the count to remind you what it is a count of! Or you can remember that in your mind. Whatever works for you.

So, whenever you read about those exotic mathematical concepts, like matrices, determinants, integrals, equations, algebra, arithmetic, you will know one thing - it is all and only about pure counts we have just talked about. There is nothing else there. For instance, a matrix is a set of counts arranged in rectangular fashion on page. But, you do not need even that rectangle. You can just imagine in your mind the same set of counts and differentiate between them in any way you want. It just happened that it was convenient to write those numbers in a rectangular grid on paper! It was just convenience.

The logic, the reasoning in the world of counted objects is separate from the logic that deals only with counts. You can investigate properties of counts only, completely independent from the real world objects they might represent count of. This is the topic of mathematics. To find what is true about counts i.e. numbers. Hence the proof. But note here, we are really interested in counts' characteristics, no matter which objects have been counted! What these characteristics can be? We can have odd numbers, or even! We can have prime counts. Some counts can be divided by others, while some not. But, the numbers are numbers, it is us who give them names to keep track of some of their properties or just we want to deal with some numbers while leaving other numbers alone!. Naming numbers is not a mathematical operation. It just help us describe, label numbers, counts, we want to deal with.

We can compare the way we can deal with numbers to the way a sculptor deals with clay. It is only the clay that he works with and nothing else. Clay! But, what clay represents when it is shaped, what sculpture represents is not about clay! The motivation how the sculptor will twist, press, mold, shape clay is outside clay's world. Same in math! The reason why we add, subtract, divide, or even select numbers to deal with, frequently are outside mathematics! The motivation can come from us buying CDs or from an economist measuring supply and demand, or police measuring speed of the car.

Now, back to sculpture again. The sculpture can represent anything. The similar thing is with numbers. In mathematics we are dealing with numbers only, the same way sculptor deals only with clay! But, if we want to interpret math results and use math in some other fields then we will have to keep track what we have counted, measured, keep track which objects are numbers, counts associated with! That would be called applied math. And, again, the numbers can represent count of many, many different things.

In developing and understanding a subject, axioms come late. Then in the formal presentations, they come early. - Rueben Hersh.

The view that mathematics is in essence derivations from axioms is backward. In fact, it's wrong. - Rueben Hersh

[ number concept, concept of a number, number, count, numbers, counts, integers, rational numbers, concept, math, mathematics ]

Saturday, February 11, 2012

Mathematical Intuition (Poincaré, Polya, Dewey), Reuben Hersh, University of New Mexico. Link to the paper.

Found an interesting paper on mathematics and intuition. Here is the summary. 

Mathematical Intuition (Poincaré, Polya, Dewey)

Reuben Hersh
University of New Mexico

Summary: Practical calculation of the limit of a sequence often violates the definition of convergence to a limit as taught in calculus. Together with examples from Euler, Polya and Poincare, this fact shows that in mathematics, as in science and in everyday life, we are often obligated to use knowledge that is derived, not rigorously or deductively, but simply by making the best use of available information — plausible reasoning. The “philosophy of mathematical practice” fits into the general framework of “warranted assertibility,” the pragmatist view of the logic of inquiry developed by John Dewey.