Mathematics
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- "Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth." – Benjamin Peirce, speaking of eiπ + 1 = 0
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[edit] Definition
Mathematics may be defined as the study of quantity, count, or amount.
This definition would suffice for most common uses pursuits of mathematics. But the idea of “number”, representing a quantity, has been generalized to include concepts that do not at first seem to be related to quantity. Vectors, imaginary numbers, “quaternions”, and “matrices”, are all considered to be generalizations of “numbers”. Additionally, mathematics has been generalized so as to encompass logic, and geometry. A more accurate, if more vague definition might be
Mathematics is the science of creating abstract or symbolic models of reality.
[edit] Mathematics as a Model
The ancient Pythagoreans believed that numbers were real entities that were somehow expressed in the mundane world. The number ‘5’, for example was considered a real thing, which might evidence itself by the fact that you had a certain quantity of sheep in the field.
The modern view is quite different. Mathematics is useful as a symbolic or abstract model of reality. The only reason you would use ‘5’ to represent your sheep in the field, is that there are certain qualities of ‘5’ that correspond to certain qualities of the sheep. One quality of ‘5’ is that it is not divisible by 3, corresponding to the fact that you could not divide your sheep into 3 equal groups.
Once we have satisfied ourselves that sheep can be represented by number, we can predict, for example, that merging a flock of 100 with a flock of 200 will result in a flock of 300. Trusting in the abstract model allows us to know that there are 300 sheep without having to tediously count them every time.
There remains a common misunderstanding that objects in the world behave a certain way because of an underlying mathematical system. Rather, a mathematical system is chosen because it behaves isomorphically to the objects. Combining flocks of sheep does not involve addition, rather the operation of addition is used to model the situation because it matches the way sheep work.
If mathematicians declared that 2 + 2 = 3, it would not cause a collapse of world money systems. Accountants would simply not use that mathematical system, since it doesn’t match the way money works.
Likewise, if we see two clouds merge into one, this does not prove that 1 + 1 = 1. Rather, it proves that the natural numbers are not applicable to counting clouds.
[edit] Mathematics in Science
One goal of science is to discover abstract models of reality, called “Laws of Nature”. These in turn become powerful tools for making predictions.
One of the greatest “Laws of Nature” ever discovered was Newton’s Law of Gravitation, which described the force between two massive objects.
Combined with his Laws of Motion, it became possible for the first time to predict all motion of heavenly objects. These laws described concepts such as motion, acceleration, force, and distance in terms of numbers, and their relations in terms of defined operations on those numbers.
[edit] Mathematical Proof and Discovery
Mathematical discovery centers on the idea of Proof. A mathematical proof does not, by itself, demonstrate some absolute fact. Rather, it begins with a set of “axioms”, and demonstrates that the fact follows logically from those axioms.
In ancient times, axioms were considered to be self-evident or obvious statements, such as “one is the smallest number”, or “there exists exactly one line through a given point, and parallel to a given line”.
Nowadays, axioms are not chosen on the basis of obviousness, but on the basis of applicability to the situation being modeled. For example, if I have five sheep and I promise someone ten sheep, then my current situation can no longer be represented in a mathematical system where “one is the smallest number”. It can, however, be represented in an alternate mathematical system, based on different axioms.
Modern science has discovered, in fact, that the system based on “there exists exactly one line through a given point, and parallel to a given line” does not model the geometry of our universe correctly! So much for obvious and self-evident axioms.
Much modern progress in mathematics is made by assuming a certain set of axioms, and seeing what can be proven. More often than not, the results do not have any immediately obvious application to reality. As with most scientific pursuits, surprising applications are discovered many years later.
[edit] The Proof of Pythagoras
The earliest mathematical proof was discovered by Pythagoras, who showed using rigorous logic that in any right-angled triangle, if x and y represent the lengths of the lines that meet at the right angle, and z is the length of the third line then:
- x2 + y2 = z2
Apparently the discovery of this proof was the cause of a massive celebration including the slaughter of 200 oxen to the gods.
But the proof of Pythagoras also proved something else that caused consternation to Pythagoras and his students. If the lengths x and y were both equal to 1 unit (inch, centimeter, light year, whatever) then the length of the third line z would be
and that 
could not be expressed as the ratio of two integers no matter how large. Thus the Pythagoreans had discovered the first irrational number, a secret that they kept hidden for years.
[edit] Mathematical Pseudoscience
Mathematical pseudoscience can take the form of applying inappropriate mathematical models to reality, or alternatively, by misusing existing models by making “proofs” using non-standard or invalid techniques.
[edit] Numerology
(Main article: Numerology) Numerology is a pseudoscience that uses numbers to make predictions. Numerology is not valuable as a predictive tool because it does not use a mathematical model that corresponds to reality. In scientific discipline, the predictions of the model must be compared against reality, and altered if they are wrong. Numerology proceeds under the untested assumption that its models are accurate.
[edit] Squaring the Circle
An ancient problem in mathematics known as “squaring the circle” involves producing a square having the same area as a given circle using certain prescribed tools (an "unmarked straightedge" and "collapsible compasses"). See main article: Squaring the Circle
Throughout history, legions of mathematicians (and cranks) have attempted various solutions to this problem. All such solutions have been erroneous, and it was proven in 1882 that the problem is impossible.
Nevertheless, many individuals continue to attack the problem, and claim solutions, using pseudoscientific verbiage to give the air of authority to their faulty reasoning.[1]
