Concept

[TODO: this page is under construction]

A concept is "a mental integration of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted."

Objective Mathematics' view of concepts is heavily influenced by the book Introduction to Objectivist Epistemology (ITOE), but with a few minor additions. The purpose of this article is to summarize the parts of ITOE most relevant to math (though by no means should it be counted as a replacement of ITOE), and to explain Objective Mathematics' additions to ITOE in detail.

A concept is a mental unit. A concept refers to an unlimited number of concretes.

With the exception of proper names, there is basically a one-to-one correspondence between the concepts that a man uses and the words that he uses. There are some exceptions to that rule. For example, in my mind, I think that "dimensional analysis" is treated as a single concept, rather than as a type of analysis which is about dimensions (though the name is apt). Conversely, in my mind the different words "squadron" and "battalion" denote the exact same concept (though I am sure that they denote different concepts for people in the military).

In German, the one-to-one correspondence between concepts and words is closer to being true, because they have more compound words than English (e.g "stop sign" is "Stoppschild"). In Chinese, the one-to-one correspondence between concepts and characters is far from being true, because often a single Chinese character is ambiguous until combined with other characters. This fact has been recognized by Chinese, which has a word ("词语") that effectively means: a grouping of one or more characters that stands for a single concept. "词语" gets translated into English as "word."

I would say something like... the English language pushes you to form concept

Most pages on the Objective Mathematics wiki are concepts.

Unit

A unit is "an existent regarded as a separate member of a [set] of two or more similar members."

Measurement

Measurement is "the identification of a quantitative relationship, by means of a standard that serves as a unit."

Definition

A definition is "a statement that identifies the nature of a concept’s units."

Any definition provides a genus, and some differentia. The genus tells us in what broad category of our mind we should look to find the concept. The differentia tell us how to distinguish the concept from the other concepts in that broad category.

For examples of definitions, look at the first sentence of most pages on the Objective Mathematics wiki.

Standard math's idea of definitions

In standard mathematics, the definition of a term is considered to be all there is to the term.

For example, take the standard math definition of a group:

Definition. A group $(G,\mu ,i,e)$ is a set $G$, a map $\mu :G\times G\rightarrow G$, a bijection $i:G\rightarrow G$, and an element $e\in G$, satisfying the following axioms

1. Associativity: $\mu (x,\mu (y,z))=\mu (\mu (x,y),z)$ for all $x,y,z\in G$.
2. Identity: $\mu (e,x)=x$ for all $x\in G$.
3. Inverse: $\mu (x,i(x))=e=\mu (i(x),x)$ for all $x\in G$.

In standard math, a group is nothing less and nothing more than what these definitions say it is. A group has exactly those properties which follow deductively from its definition, and no other properties.

In contrast, the properties of objects in real life are identified rather than deduced from axioms. Consequently, there is always the possibility that a real-life object has properties which have not yet been identified---that some totally novel property of the object could be discovered. However, anything that is discovered about (the standard math notion of) a group, though it may have been previously unknown, is not novel in the same way: it is necessarily just a deductive consequence of the group axioms.

For the concepts of standard mathematics, there are no borderline cases or grey areas. Given some mathematical structure (in standard mathematics), either it is a group, or it isn't a group. In real life, though, there are borderline cases. There are objects whose measurements don't clearly fall inside the range of a given concept, but also don't clearly fall outside the range of that concept. For example, is a feminine hermaphrodite a woman? I imagine that there is not always a clear answer to that question.

AR describes what is wrong with this idea of definitions:

A definition is not a description; it implies, but does not mention all the characteristics of a concept’s units. If a definition were to list all the characteristics, it would defeat its own purpose: it would provide an indiscriminate, undifferentiated and, in effect, pre-conceptual conglomeration of characteristics which would not serve to distinguish the units from all other existents, nor the concept from all other concepts.

Specification [TODO]

In certain contexts, something more like standard mathematics' idea of definitions is what we need.

For example, if one is programming computer, then one has to specify exactly what all one's "concepts" (i.e. classes or types) mean. Computers are dumb, so there is no room to be vague, or to leave out anything non-essential. To define a class in, say, C++, one must precisely specify all of its attributes, down to the most trivial details.

Another example of a context in which one needs a "definition" which includes more details than what Objectivism requires (but fewer details than what a computer program requires) is: legal definitions. To identify the nature of the units of the concept "man," it suffices to say that man is a rational animal. But to legally define "man," one has to specify many more things, like at what---if any---level of brain damage (and by what standard of brain damage) shall someone cease to be counted as a man, at which precise time shall a fetus be counted as having begun its life as a man, etc.

Lastly, and most importantly for Objective Mathematics, there are many situations where precision seems to require us to specify more than just those characteristics of a concept's units which are essential. For an example of such a situation, consider the concept of a prime number. The definition of prime number is: a natural number which cannot be factored nontrivially into natural numbers. That definition states the essential properties of primes. However, from that definition, it is not clear how one should deal with the borderline case of 1: does $1=1\times 1$ count as a "nontrivial" factorization or not? In some antiquated definitions of primes, 1 is counted as a prime, and indeed, I think that there is not an essential reason why it is wrong to count 1 as a prime. It is, however, generally more convenient to count 1 as a non-prime. For example, if 1 were to be counted as a prime, then the fundamental theorem of arithmetic

Fundamental Theorem of Arithmetic. Any natural number admits a unique decomposition into primes.

would be more complicated to state, for the decomposition into primes would not be not unique. If one wants to use a theorem like the above, then one's concept of prime must agree with that of the person who stated the theorem, even down to some minute and trivial details like whether 1 is a prime. So we see that beyond just the "definition" of a prime, some extra specification of details can be useful in math.

In view of situations like those stated in these three examples, I reserve the word specification to mean: a definition which gives more details than required by the rules of concept formation, but rather gives just as many details as are required to meet some other end.

Notion

A notion is a compound concept. A notion is like a concept, in the sense that it subsumes and includes an unlimited number of concretes. A notion is unlike a concept, in the sense that it might be made of other concepts.

[TODO maybe I should rename this to "phrase"....]

Examples:

• 3 is a concept, but 16654 is a notion. In general, any number is a notion.
• "Fox" is a concept, but "the quick brown fox" is a notion. In general, any noun phrase is a notion.
• "To jump" is a concept, but "to jump off the gangplank with one's hands tied behind one's back" is a notion.

If a concept does not refer to any concretes, then it was formed invalidly. By contrast, one can use valid concepts to validly form a notion that does not refer to any concretes in reality. For example, "the talking purple platypus" does not refer to anything in reality, but it nonetheless makes sense, and "the," "talking," "purple," and "platypus" are all valid concepts. For another example, $9^{9^{9^{9^{9^{9^{9^{9}}}}}}}$is too large to refer to anything in reality, but it is clear what it means, and 9 and exponentiation are both valid concepts.

I reserve the right to be sloppy about the distinction between notion and concept, in situations where I think it doesn't matter.

Proposition

A proposition is a basic identification of a fact of reality. It is "the form in which we make conceptual identifications."