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Posts Tagged ‘**lemma**’

**Idiosyncrasy in “conjectures”; (Dec. 21, 2009)**

Idiosyncrasy or **cultural bias** relates to “common sense” behavior (for example, preferential priorities in choices of values, belief systems, and daily habits…) is not restricted among different societies: it can be found within one society, even within what can be defined as “homogeneous restricted communities” ethnically, religiously, common language, gender groups, or professional disciplines.

**Most disciplines have mushroomed into cults**.

A cult is any organization that creates its own **nomenclature** and definition of terms to be distinguished from the other cults in order to acquiring recognition as a “**professional entity**” or independent disciplines that should benefit from laws of special minorities (when mainly it is a matter of generating profit or doing business as usual).

These cults want to owe the non-initiated into believing that they have serious well-developed methods or excellent comprehension of a restricted area in sciences. The initiated on multidisciplinary knowledge recognize that the methods of any cult are old and even far less precise or developed; that the terms are not new and there are already analogous terms in other disciplines that are more accurate and far better defined.

Countless experiments have demonstrated various kinds of idiosyncrasies. This article is oriented toward “cult” kinds of orders, organization, and professional discipline. My first post is targeting the **order of mathematicians**; the next article will focus on experiments.

Mathematics, meaning “sure study” (wisekunde), has no reliable historical documentation. Most of mathematical concepts were written many decades or centuries after they were “floating around” among mathematicians.

Mathematics is confusing with its array of nomenclature. What are the differences among axiom, proposition, lemma, postulate, or conjecture? What are the differences among the terms, theorem, questions, problems, hypothesis, corollary, and again conjecture? For example, **personally, I feel that axiom is mostly recurrent in geometry, lemma in probability, hypothesis in analytical procedures, and conjecture in algebraic deductive reasoning.**

Hypothesis is in desuetude in mathematics. For example, Newton said “I am not making a hypothesis”.

**Socrates** made fun of this term by explaining how it was understood “I designate hypothesis what people doing geometry use to treating a question. For example, when asked for their “expert opinion” they reply: “I still cannot confirm but I think that if I have a viable hypothesis for this problem and if it is the following hypothesis… then I think that we may draw a conclusion. If we have another hypothesis then another conclusion is more valid.”

**Plato** said: “As long as mathematics start from hypothesis instead of facts then we do not think that they have true comprehension, since they are not going back to fundamentals”

Hypothesis is still the main term used in experimental research. Theoretically, an experiment is not meant to accept a hypothesis as true or valid, but simply “Not to reject it” if the relationships among the manipulated variables are “statistically significant” to a pre-determined level, usually 5% in random errors.

Many pragmatic scientific researchers don’t care about the fine details in theoretical mathematical concepts and tend to adopt a hypothesis that was not rejected as law. This is one case of idiosyncrasy when the researcher wants badly the “non-rejected” hypothesis to represent his view. Generally, an honest experimenter has to repeat the experiment or encourage someone else to generalize the results by studying more variables.

Conjecture means (throwing in together) and can be translated as conclusion or deduction; basically, it is an opinion or supposition based on insufficient proofs.

In the last century, conjectures were exposed in writing as promptly as possible instead of keeping them floating ideas, concepts, or probable theorems. This new behavior of writing conjectures was given the rationale that “plausible reasoning” is a set of suppositions thrown around as questions mathematicians guess they have answers to them, but are unable to demonstrate temporarily.

The term conjecture has been used so freely in the last decades that **Andre Weil** warned that “current mathematicians use the term conjecture when they fail after a few attempts to verify a concept, even if the problem is of no importance.” **David Kazhdan** ironically warned that this practice of enunciating conjectures might turn out like a 5-year Soviet plan.

At first, a set of conjectures was meant to be the basic structure for a theorem or precise assertions that were temporarily used in the trading of logical discussions. Thus, conjectures permit the construction of rigorous deductions that are accessible to direct testing of their validity. A conjecture was a “research program” that move ahead in order to foresee the explored domain.

Consequently, conjecture is kind of extending a name and an address to a set of suppositions and analogies for a concept, long before tools and methods are created to approach directly the problem.

A “Problem” designates a mental task submitted to the audience or targeted for research or project; usually, the set of problems lead to demonstrating a general theorem. Many problems are in fact conjectures such as the problem of twin primary numbers that consists of proving the existence of an infinity of coupled numbers such that p-q = 2.

One of the explanations for using freely the term conjecture is the modern facility of mathematicians of **discriminating aspects of uncertainty at the theoretical level.** It is an acquired habit, an idiosyncrasy. Thus, for a mathematician to state a conjecture he must have solved many particular cases and recognize that a research program is needed to developing special tools for demonstrating the conjecture. **This is a tough restriction in this age where time is of essence among millions of mathematicians competing fo**r prizes.