Beginnings 1

1 + 1 :: Int

2

The perception of objects: separate or together, one or many, close or far away. Higher life on this planet would seem to have an innate, intuitive grasp of such contrasting conditions. We humans have language, and this brings logic and symbolism to formalize intuition. Consider the statement

⊕ The ⟹ arrow is the mathematical logic symbol for implies, as when one thing implies another thing. Or, roughly, “if you say this, then you are also saying this.” We’ll describe all the math logic symbols as we go.

⇲    together \(\implies\) not alone, not separate

A statement is also called a proposition. The first thing to know about propositions is they are either true or false. As in, We propose this to be true, but it might be false. Reading the statement backwards

⇲    not alone, not separate \(\implies\) together,

we see a simple “if not this, then that” situation. Both version are statements, or propositions we can decide to be true or not. But not every utterance is a statement. For example

Out of these three sentences only #1 is a proposition, a statement that can be resolved to true or false. #2 is a command, and #3 is a question. Of course #2 and #3 can be answered with yes or no, or good or bad, but not with true or false.¹¹ There is a story of the famous physicist Paul Dirac who, finishing a guest lecture, asked the student audience if there were any questions. A student said, “I don’t understand how you got…” But Dirac seemingly ignored him. A professor then asked Dirac if he would answer the question. Dirac replied, Was there a question? I only heard a statement.

Our together \(\implies\) not alone, not separate makes sense, seems true. But could there be lurking somewhere a situation where together does not really imply or mean not alone and not separate. What about riding on a city transit bus? Yes, you can technically be together with all the other passengers, but still be quite alone and separate from all the people you do not know. The takeaway is we have a two-part process: First, the statement, then whether it is true or false.

Aristotle, who lived in third century BC and was Plato’s best-known student, based his logic on three rules

1. Identity: A thing is itself, or, \(A \; \text{is} \; A\)
2. Excluded middle: A proposition is either true or false, or, \(A \lor \neg \,A\)
3. Contradiction: A proposition cannot be both true and false at the same time.

These seem rather—obvious, to the point of being silly. However, logicians and mathematicians must be formal and thorough about it. Let’s analyze each in turn.

➝ \(A \; \text{is} \; A\)
is probably the most pedantic and even tedious seeming. But in the quest to force nebulous intuitive phenomena to conform to logical thought and speech, we must at least use symbols consistently. Again, it seems obvious that a dog is a dog, but we cannot have dogs randomly be cats. If Shakespeare’s Juliet declares

What’s in a name? That which we call a rose, by any other word would smell as sweet.

we still need whatever the rose is now called to stay that name and not have different names coming in and out of usage at random. This speaks to a need for consistency.

Aristotle said

If … one were to say that [a] word has an infinite number of meanings, obviously reasoning would be impossible; for not to have one meaning is to have no meaning, and if words have no meaning our reasoning with one another, and indeed with ourselves, has been annihilated; for it is impossible to think of anything if we do not think of one thing…

Again, language is forcing us to exactness. Aristotle also employs symbols such as X and A to generalize his argument

if “man” means X, then if A is a man X will be what “being a man” means.

Again we see implication²² …and more generally, an implication consists of a pair of sentences separated by the \(\implies\) operator. Symbolically, \(p\) implies \(q\) or \((p \implies q)\). The sentence to the left of the \(\implies\) operator is called the antecedent, and the sentence to the right is called the consequent. where if we have a meaning X given for what a man is, then if an individual A is considered as man, then we must stay with the meaning X of what a man is. Man means X. A is a man. Hence, A means X.

The British mathematician George Boole emphasized the fact that with language we use words, names, symbols, and these are in truth made-up, markings on paper or blackboards, sounds, pronunciations, but they must have set shapes, sounds, and meanings agreed upon by all users.

[What does Boole mean with Nothing “0” and Universe “1”: He observes that the only two numbers that satisfy xx = x are 0 and 1. He then observes that 0 represents “Nothing” while “1” represents the “Universe” (of discourse). Is this some forcing logic? After all neg times neg is forced by (50 - 2)(50 - 2) = 48 x 48, i.e., we follow one trail taught in grade school to “figure out” 48 x 48, but is (50 - 2)(50 - 2) a forcing logic?]

➝ \(A \lor \neg\, A\) (A or not A)
is quite prevalent in everyday life, even if we don’t realize it. Whenever we hear a statement such as If you don’t clean your room, you’ll be grounded for the rest of the week! Here the parent has created an excluded middle: Clean your room, or be grounded—excluding any other possibilities. Harsh, but common in everyday life. Of course the usual rebuttal is neither-nor or—both. We see both when someone is offered two appealing choices—and doesn’t want to be restricted to one or the other, no both, please.

There cannot be a world where we

Abstraction and symbolism have taken humanity into a modern science and technology landscape that, as far as we know, is unique in the known universe.

things that were together, not just singular by themselves? We would see more than one thing—perhaps similar, perhaps close to one another—and the notion of relatedness, togetherness, clumpiness would come to us. Human language Some vague idea, if not language, dealing with amount rose from how big the clump of things we mentally clumped together seemed.

The general phenomenon of singular and plural, one or more, seemed to beg the question if more, then how many more? And so the size or degree of a clump of things means something and should be considered.

Along with clumpiness, groupings, stuff together, we probably also were aware of the passing, the demarcation of time, such as the seasons cycling. And shapes could be categorized as well.⁴⁴ Geometry. But let’s stick with clumpiness for a while.

Communication, semantics of these clump details evolved. A couple, a pair, a herd, a school, a few, many, lots, lots and lots, lots and lots and lots. And today terms like cardinality or degrees of multiplicity offer a high level of abstraction of one or many.

All of this points to a very unique thing about humans, namely, our high levels of cognizance, awareness, our ability to note our intuitive awareness and build on it. Of course other critters are able to distinguish between one and many. But do they put these notions, these impulses into words—and then build these ideas into higher and more exacting and more abstract ideas? Of course humans experience life at an intuitive level, but unlike animals, we take the intuitive and give it logical structure, we codify these notions, ideas, intuitive hunches. We, far more than any other species take basic phenomena and create abstract representative mental structures.