Sometimes, in math classes, we are asked to give justification for our mathematical assertions. We say that mathematical statement X is true because Y is true. However, I don't know if "because" is the right word to use. Mathematical objects and statements, in my view at least, are not located in time or space, so they can't cause anything. So, why do people talk of "because" in math? Is it that they really mean "implies"? But I don't think "implies" is the right word to use, either. Every true mathematical statement implies every true mathematical statement, but, for example, we wouldn't say that the fundamental theorem of algebra is true because 3+4=7. So, then, my question really is, has any philosopher or mathematician formalized a notion of "because" in mathematics?
-
2See SEP, Mathematical explanations within mathematics. The most common use of "because" is to point out a general result that makes an initially surprising phenomenon in some example part of a general pattern. This is not part of the formal logical model of mathematics that only captures logical justification, which is why there is no correlate for it in formal theories.– ConifoldJun 21, 2022 at 19:07
-
4Formalizing the notion of "because" is arguably the motivation for relevance logic.– Jim BelkJun 22, 2022 at 3:36
-
8In layman terms, what you are really saying when you say "X is true because Y and Z are" is in fact "I can say X is true because I know Y and Z are". In that sense the cause effect relationship is correct. You had the correct intuition about math, if only because math propositions can often be reversed: "this triangle is right because a^2 = b^2+c^2" <-> "a^2 = b^2+c^2 because this triangle is right". One term is not the cause of the other, both terms are synonymous.– armandJun 22, 2022 at 9:21
-
2@armand I think this should be an answer.– Jordi VermeulenJun 22, 2022 at 9:31
-
10"Every true mathematical statement implies every true mathematical statement" is only correct in a very limited sense and only in classical logic. Proof theory usually adopts a much less blunt and more useful stance: P implies Q means that a proof of P can be converted into a proof of Q.– Mark DominusJun 22, 2022 at 14:03
9 Answers
It seems like the type of "because" statements you are talking about can be equated to "is implied by" (or perhaps more precisely "is derivable from"). For example 3+4=7 is implied by the most fundamental axioms of arithmetics (e.g. Peano), but not the other way around, because 3+4=7 is not given as an axiom. However, I see that "because" is not really the most rigorous word for that case, so I believe it is just people being a bit liberal in how they word questions.
That said, there are notions of "because" in applied mathematics, specially in the field of statistics, under the general field of causal inference. A few embodiments of it are randomized controlled trials, counterfactual conditionals, and Judea Pearl's causal graphs.
-
2I agree. I'd add that in the study of foundations of mathematics there is a lot of work to show which sets of axioms imply which other sets of axioms. It turns out there are many different ways to build a foundation of mathematics and it is quite a complex problem to know the relative strength of these different foundations. However, for most relevant mathematical questions, the various foundations of sufficient strength imply the same theorems. This is why in contemporary math, foundations are often ignored or treated as an afterthought.– Avi CJun 21, 2022 at 17:21
-
6Rather than "is implied by," I think you mean, "is derivable from." i.e., 3+4=7 is derivable from the Peano axioms, but not vice versa. In mathematics, any two true statements imply each other, so that's probably not what you mean. Jun 22, 2022 at 0:10
-
1I agree: "I'm using an umbrella, because it's raining" is equivalent to "using an umbrella implies it it raining". Actually it means: "Whenever it is raining I'll be using an umbrella" (I may still use an umbrella when it's not raining) which is stronger than the original because sentence.– U. WindlJun 22, 2022 at 7:41
-
I'm not sure that the implies vs derived distinction carries any philosophical weight. In a proof system with standard meta-theoretic properties, if there is a proof of "P implies Q", there is a proof which assumes P and proves Q (i.e., derives Q from P).– Avi CJun 22, 2022 at 15:47
The answer is actually quite trivial once you understand formal proofs. Take for example this Fitch-style system. Each line in a proof in this system must be deduced from preceding lines. If you skip steps, it would mean omitting some lines. 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, ...). For example, in a formal proof you may have this subproof:
...[1]
A.
where there are multiple lines in "...[1]" that are only used to deduce "A". In that case, an informal proof might present those steps in a manner that essentially has the following structure:
A, because:
...[1]
This should be understood as a top-down presentation, whereas the earlier formal subproof is a bottom-up presentation. 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.
So in mathematical writing, "A, because B." is simply no different from "B. Thus A.". Other remarks about philosophy are simply irrelevant to the usage of "because" in common mathematical writing.
-
From this it should also be obvious why it is not the same as an implication, since "B. Thus A." is not the same as "B ⇒ A.". Jun 22, 2022 at 9:05
I believe “because” in the sense you want enters at the foundational level of mathematics, as in how one views mathematical objects beyond the formalism.
Why does 3+4=7? Because it somehow represents abstract truths (platonism)? Because it is necessarily so from our innate concepts (Kant)? Because we’ve provided a positive proof (intuitionism/constructivism)? Because it serves a purpose while being false (fictionalism)? Because it is a biological hardwiring (Lakoff)? Because it serves the same structural role as other structures (some type of structuralism)?
I think you’re asking for the deepest because we can offer. And we can’t definitively answer that.
-
Would it make sense to call this notion of "because" the metaphysical grounding relation for mathematical truth?– Avi CJun 21, 2022 at 17:56
-
1@AviC maybe I can get away without necessarily involving metaphysics, that way I can include things like Lakoff who might deny metaphysics in his explanations? But also I think fictionalism may not appreciate being necessarily metaphysically implicated. My answer is pretty basic and regurgitative, but maybe OP hasn’t heard of this stuff before. (I know I hadn’t until very late)– J KusinJun 21, 2022 at 19:54
-
1Certainly your last short and succinct paragraph is not repetitive/regurgitative! Jun 22, 2022 at 2:00
-
@DoubleKnot genuinely not sure! If 10 different answers are provided for the same question, am I being metaphysical by saying there is no definitive answer?– J KusinJun 22, 2022 at 5:23
-
That may be the deepest of the deepest metaphysical question lol... Look at Platonic heavens without causality as assumed by OP, all logic forms of material implications can be reduced to a disjunctive normal form concatenated by "or"s... Jun 22, 2022 at 5:39
You can rephrase "x=1 because x+1=2" to "x+1=2 therefore x=1" or
x+1=2
∴x=1
Yes, almost(?) every mathematical proof has one or more "because" in it, but (according to my experience) usually the word "since" (which, you will certainly agree, is equivalent to "because") or "given that" would be preferred.
Logically, "A because B" is equivalent to "B implies A", and the general structure of mathematical proofs is most frequently of the form "premise => conclusion", so words meaning implication, like "thus", "hence", "whence", "therefore" will me more frequent than "since" or "because".
Also, "because"/"since" will often be used to make the reading of the proof more easy by recalling or restating things that had already been stated previously, and therefore might even be omitted in a denser style of proof.
You should have no difficulty to check these claims by scanning through mathematical papers as you can find abundantly, for example, on https://arxiv.org/list/math/recent. But if required, I'll be glad to provide concrete examples/references.
Because means either:
by reason of
or by cause of (as in cause and effect, a temporal relation).
We can formalise the first notion by qualifying propositional logic into modal logic with the modality operator, it necessarily follows as opposed to it possibly follows or is impossible.
And we can formalise the second by temporalising our logic. This was first done comprehensively by Ibn Sina (Avicenna) because he believed most categorical propositions were not true without temporal qualifiers. There were precursors to Ibn Sina in stoic logic, for example, in the writings of Theophrastus.
Yes.
The symbol is:
∵
which is an upside down "therefore".
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).
There may be value in a formalized meaning that is similar. 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.
This also lets us break the two terms apart when it suits us. In your example, 3+4=7 is necessary for the fundamental theorems of algebra to hold true, but it is not sufficient. Likewise, the rules of Peano Arithmetic (the rules of arithmetic you are probably most familiar with) are sufficient to show 3+4=7, but they are not necessary. Weaker arithmetic, such as Presburger Arithmetic are also sufficient.
Neither of these quite captures causality the way "because" does, however it can do so in context. Often we find ourselves with an implication where the antecedent is a "higher order" than the consequent. In those situations, it's reasonable to say that the implication acts like "because." But it isn't necessary. 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 most formal logic systems, every individual statement is independently provably true, provably false, or undecidable, based upon the system's fundamental axioms. 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. 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.
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. 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. 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.
