Why Math Needs Logic, But Logic Doesn't Need Math
C.P. van der Velde.
[First website version 31-03-2025]
1.
Introduction
The relationship between mathematics and logic remains a quiet cornerstone of thought,
often overlooked but unshakably vital.
Math - numbers, equations, proofs - seems to rule precision,
yet it leans on logic, the art of reasoning itself. Flip it around, and logic stands alone,
needing no numbers to thrive. This isn't a petty turf war - it's about what makes each tick
and why one depends while the other doesn't.
Picture math as a grand building and logic as its foundation: one can't rise without the other,
but the ground holds firm solo. Here's why.
2.
Math: Built on Logic's Shoulders
Math needs logic like a house needs a base. Every equation, every theorem, every proof,
rests on rules of reasoning.
Take "
2 + 2 = 4". Seems simple, but peel it back: it's true because "
if two things join two more, four result - necessarily": a logical step.
Facts (premises) lead to answers (conclusions). Example: "
All even numbers divide by 2.
4's even, so 4 divides by 2." That's "
if-then" - math's backbone.
Geometry's proofs - like proving triangles congruent - string "
if-then" chains:
if side A equals side B, and angle C matches D, then the shapes align.
Even advanced fields like calculus or algebra rely on axioms, basic truths logic validates. Try defining "
1" without reasoning "
one thing differs from none" - logic's there first, framing it all.
Math
always uses logic, and without it's glue
- say, modus ponens (if A implies B, and A's true, B follows) - math's claims collapse.
3.
Logic: The Standalone Root
Logic, though, doesn't need math to breathe. It makes out the order and coherence in our thinking,
existing before numbers enter the scene.
Think everyday reasoning - natural logic. "
If it's raining, I'll stay in. It's raining, so I'm staying.
" Fact to result, no numbers. That's logic - pure, no digits required. It deals in statements
- true, false, linked by rules like "
and", "
or", "
not" - and spins out conclusions by "
if-then" implications. Aristotle's syllogisms - "
all men are mortal, Socrates is a man, so mortal
" - work without counting a thing. Like: "
All dogs have ears. Spot's a dog, so Spot has ears."
Formal logic still rules this way: a single distinction ("
this, not that") births
the entire 'machinery' of logic, an infinite web of possible combinations and their implications,
completely indepentent of time and space.
4.
Important concepts in Logic
Logic has many distinctions and criteria that are quite useful - actually essential -
in daily thinking and judgement, all independent of numbers or calculations.
4.1.
Sufficient vs. Necessary
Sufficient is "
enough" - "
If I study, I'll pass"
means studying is one condition that suffices to succeed.
Necessary is "
must-have" - "
I pass only if I study" means studying's required, but might not be sufficient as a condition.
Mix-ups confuse: "
If it rains, I'm wet" (sufficient) versus "
I'm wet only if it rains"
(necessary - shower says no!).
There is hoewever an interesting relation: a sufficient condition comprises all necessary conditions for a
conclusion.
4.2.
Syntax vs. Semantics
Syntax is the form - "
if A, then B." A clean logical structure, but is it true?
That may be dependent on contextual information.
Semantics tests the content through interpretation of the statement in a certain context.
If then it appears that A never occurs without B, you can rightfully conclude "
if A, then B".
There's a funny relation here. We can talk about a "crazy logic", a "weird logic" etc.,
not meaning that logic as such can be crazy or weird. The argument put forward
may have the logically correct syntactic form of a reasoning - but it's semantic
content
is considered, well, rather fiddle-faddle.
4.3.
Truth vs. Validity
Truth is concerned with states of affairs in a certain domain. Like, in the domain
of dogs, Spot barks (yes) or doesn't (false). We're keeping it basic - yes/no - but fancier reasoning adds
modal qualifiers, like possibly, probably, unsure. In any way, truth value is dependent on certain
contexts, be it even by arbirary
assigmnent like, "
well, I just belkieve it's true".
Validity is tougher - it means "
always true" if the form holds. "
If Spot barks, he's a dog and/or a cat" seems silly, but never fails if you think about it.
It only has a weaker conclusion then it's premisse.
In contrast to truth, validity holds independent of specific contexts interpretations or assignments.
On a more abstract level, "
If A, then A and/or B" will be always true, in any corner of the universe.
4.4.
Inconsistency
Inconsistency if the exclusion of validity. "
A and B and A means not B" - like "
I'm happy and sad and happy means I'm not sad" - common mess in sloppy thinking.
4.5.
Contingency
Contingency means a rule is neither valid nor inconsist, like "if A, then B"
when there is sufficient ground - facts or other rules - to call it valid or busted.
4.6.
Sound Reasoning
A reasoning is sound if the conclusion logically follows from a set of valid rules,
and a set of true facts. "All dogs bark, Spot's a dog, so Spot barks" is sound
- provided no dogs ever lacks the ability to bark.
5.
Why the Dependency Runs One Way
Why can't math stand alone? Because it's a system within logic's domain.
Numbers and operations - addition, multiplication - assume logical consistency: "
2 + 3 = 5
" holds only if equals means equals, a logical rule. Try math without it: "
2 + 2 might be 5" if "
might" replaces "
is", and chaos reigns. Proofs falter - Fermat's Last Theorem needs "
if A, then B" to chain steps, not random leaps. Logic, meanwhile, shrugs off math's tools. "
If the sun rises, it's day" doesn't care about 24 hours or angles - it's true by reasoning alone.
Math quantifies; logic qualifies, and the latter's enough by itself.
6.
Examples That Show It
Picture a courtroom. "
If the suspect wasn't at the crime scene, he's innocent; he was elsewhere, so excused" - logic, no math.
Now, a budget: "
$20 plus $80 is $100" - math, but only if "
plus" means "
and" logically.
AI machines use logic to parse posts on social media - "
if this word, that intent" -
before math crunches sentiment scores. Even infinity, math's darling, bows to logic: "
if numbers never end, then .." starts with reasoning, not digits. Math builds towers;
logic digs the earth they stand on.
7.
The Bigger Picture
This one-way street matters. Math's power - solving orbits, coding apps -
rides logic's clarity, giving us planes and phones.
But strip logic away, and math's just symbols with no spine.
Math needs logic's form, facts, and conditions - lose it, 2 plus 2 flops.
Logic's independence shines brighter: it predicts, judges, concludes - court verdicts, daily choices -
without needing a calculator.
Nowadays, as tech leans on both, logic's quiet primacy holds. A computer's "
if-then" code echoes it;
a brain's "
this, so that" mirrors it. Math's logic-bound, it needs logic to mean anything at all.
Logic needs nothing but itself to know what any information entails.
It works free and limitless with it's laws and relations - while math's optional.