![]() So, to represent very large or very small numbers concisely, we use the standard notation. It does not, however, make sense to say that any statement is true because some non-axiomatic statement is true, since the truth or falsity of both statements would independently flow from the axioms, and they would in no way affect the truth or falsity of each other.A standard notation is a form of writing a given number, an equation, or an expression in a form that follows certain rules.įor example, 4.5 billion years is written as 4,500,000,000 yearsĪs you can see here, writing a large number like 4.5 billion in its number form is not just ambiguous but also time-consuming and there are chances that we may write a few zeros less or more while writing in the number form. If a certain statement can be proven true in a system with twenty axioms, and could be proven just as well with five of them omitted, but omitting any of the other fifteen would render the statement false or undecidable, then it would make sense to say that the statement is true because the fifteen necessary axioms are true. ![]() The only sense in which the notion of causality would be meaningful in such systems would be in deciding which axioms would be necessary to prove which statements. From a practical matter, it may be useful to recognize categories of statements for which proofs of true or falsity might exist, but for which no such proofs are known to exist, but such categories would not be part of the systems themselves. If a statement is provably true or provably false in some system, that would be an inherent property of the statement and the system, and would in no way be affected by anyone's ability or inability to prove that the statement is true or false. In most formal logic systems, every individual statement is independently provably true, provably false, or undecidable, based upon the system's fundamental axioms. While mathematicians may find it useful to think in terms of "because," the math itself typically benefits from other phrasings such as "necessary and sufficient," or through implications that are structured in a way to suggest causality. In those situations, it's reasonable to say that the implication acts like "because." But it isn't necessary. Often we find ourselves with an implication where the antecedent is a "higher order" than the consequent. Neither of these quite captures causality the way "because" does, however it can do so in context. Weaker arithmetic, such as Presburger Arithmetic are also sufficient. Likewise, the rules of Peano Arithmetic (the rules of arithmetic you are probably most familiar with)Īre sufficient to show 3+4=7, but they are not necessary. In your example, 3+4=7 is necessary for the fundamental theorems of algebra to hold true, but it is not sufficient. This also lets us break the two terms apart when it suits us. I would argue that that meaning is typically seen broken into two parts, "necessary and sufficient." These are terms that are typically tied to entailment, and "entails" is a very carefully formalized word. There may be value in a formalized meaning that is similar. I don't think "because" has a meaning in mathematics akin to the English word "beacuse," simply because when we need the meaning of the English word, we just use it in English (or any equivalent phrasing in other languages). Other remarks about philosophy are simply irrelevant to the usage of "because" in common mathematical writing. So in mathematical writing, "A, because B." is simply no different from "B. Note that both kinds of presentations can be made equally formal, but conventionally logicians have defined abstract formal systems to be bottom-up for easy analysis rather than for easy practical use. This should be understood as a top-down presentation, whereas the earlier formal subproof is a bottom-up presentation. In that case, an informal proof might present those steps in a manner that essentially has the following structure: Where there are multiple lines in "." that are only used to deduce "A". For example, in a formal proof you may have this subproof: But informal proofs often not only skip logically simple steps but also present the remaining steps in a different order from formal proofs in most abstract formal systems (whether Fitch-style, sequent-style, Hilbert-style. If you skip steps, it would mean omitting some lines. Each line in a proof in this system must be deduced from preceding lines. Take for example this Fitch-style system. The answer is actually quite trivial once you understand formal proofs.
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