Team. Many among us are searching for an "absolute" truth. In an understanding of this lies certainty. From which, we can distill the reliable consequences of any action. Hence, many find solace in "faith-based" readings. It is their hope that they might further unravel and understand the dynamic between spiritual actions and reactions. So, they know that certain behaviors elicit a predetermined set of responses. Such knowledge can potentially increase the "spiritual" stability in one's life.
It is this same desire for "predictability" and "stability" in the physical world that makes some pursue the field of mathematics. For seemingly, when one adds a pair of numbers "a" and "b", the result is consistently "c". This obviously apparent and timeless rule can never be violated.
And, it is through the historical investigation of such rules: +, -, *, and /, that a myriad of mathematical disciplines and subdisciplines have been established. This includes algebra, topology, calculus, and the theory of graphs. These subjects and their subfields hold "truths" which the modern mathematician sees as "self-evident" much like those inalienable "freedoms" of life, liberty, and the pursuit of happiness. Yet, are they? And, if not "self-evident", are they identically true.
And, if one might establish that a "fundamental" flaw exists in this "timeless" mathematical reasoning, how will the modern mathematician proceed? Will he persist in his "pollyannic" view of his discipline like a cogitating ostrich. Or, will he accept that centuries of reasoning performed by men which he deifies and holds as unquestionable, such a Descartes, Euler, Fermat, and Gauss, might be horribly flawed and incomplete?
One such observation concerning the sets of established numbers has been made on numerous occasions. And that is this, the set of "irrational" numbers simply is that, "irrational". The notion that another "whole" or "floating-point" number cannot be seen as a ratio of "whole" numbers is incredibly "counter-intuitive". In fact, it grates against "natural" mathematic instincts so much that one must consider such a mathematical proposition as "highly" suspect.
And, before one "raises" his mathematical hackles, consider this simple construction, an infinite "ten" or 100....0. This one followed by an infinite expansion of zeroes, inf(10), is a "permissible" whole number. Although it has an abbreviated description, its complete expansion could never be rendered.
Now, take any "irrational" number with a "non-terminating" decimal such as pi or e and multiply them by inf(10).
So, pi * inf(10) is an element of the set of natural numbers. Let us call it, PI. So, pi = PI/inf(10). And, pi is a "rationale" value formed from non-terminating whole numbers. The same is true of "e" or any other scalar which has previously been presented as "irrational". And, hence, any postulate, theorem, or worse yet, fundamental axiom, which has been established in the disciplines of mathematics since the establishment of the delusionary notion that number could actually be "irrational" is potentially flawed.
So, "certainty" does not exist in mathematics, or for that fact, computation itself, in that it is built upon man's grossly limited, inherently flawed, and highly fallible capacity for reasoning.
And, it can be said that, with the introduction of the notion of the "irrational" number in the fifth century by the Greek mathematician Hippasus of Metapontum, mathematics itself has entered the realm of the "irrational". And, it should be said that as of 2021 A.D., it has not left.
irrational [ ih-rash-uh-nl ]
adjective
- without the faculty of reason; deprived of reason.
- without or deprived of normal mental clarity or sound judgment.
- not in accordance with reason; utterly illogical: irrational arguments.
- not endowed with the faculty of reason: irrational animals.
Mathematics.
- (of a number) not capable of being expressed exactly as a ratio of two integers.
- (of a function) not capable of being expressed exactly as a ratio of two polynomials.
Algebra. (of an equation) having an unknown under a radical sign or, alternately, with a fractional exponent.
Greek and Latin Prosody.
- of or relating to a substitution in the normal metrical pattern, especially a long syllable for a short one.
- noting a foot or meter containing such a substitution.
noun
Mathematics. irrational number.
