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Questionsaboutstuff's avatar

If you had to answer why the axioms of logic and mathematics are the way they are, what would you say?

Asked by Questionsaboutstuff (233 points ) December 29th, 2012

First how did we have intuition of a set of axioms that has served us well and if we believe they are true why are they that way?

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15 Answers

dxs's avatar

I’m a bit confused at what you’re asking, but here is how I understand it: “Why do we have axioms and why are they the way they are?”
If this is what you are asking, then I think that we have axioms to better understand the world that we live in. They are the way that they are because they are truths about the world, or truths we hold to be self-evident, to quote the Declaration of Independence.

Questionsaboutstuff's avatar

@dxs
Not why do we have axioms, but why did we have intuition for logic such as if A=B and B=C then A=C and all the axioms in mathematics.

The second part of the question is why is the logic we have true, so why doesnt 1+1=6

pleiades's avatar

With mathematics we can account for all things that exist in the physical realm because we are merely counting and subtracting of things, substance, patterns. This is why it will be very hard to determine whether the 4th, 5th, 6th dimensions and up are made up of stuff, substance , patterns and etc. We cannot yet comprehend the actuality/realities of those that Einstein theorizes about.

So I hope that puts in perspective what we can infer about an axiom. The basis of logic is for us to all acknowledge what we are all seeing, hearing, feeling, smelling and tasting.

1+1=2 because 1 represents 1 thing only.

wildpotato's avatar

GQ. In short: we take in perceptual appearance (the “intuition” you speak of) and then spit the same information back out in our explanations of it (axioms), because our faculty of understanding the world operates in terms of formal logical necessity, and it imposes this presumption of formal logical necessity on what it perceives. To flesh out my assertion, I offer this quote from the philosopher H.S. Harris:

“The ‘experience that Understanding makes’ of its syllogistic connection with the inward nature of things, is an experience of the formal necessity that it imposes upon its own ‘ideas.’ Its science is ultimately all a matter of what Hume called ‘the relations of ideas’ (or of what Hegel calls formal, and sometimes “external,” necessity). There is no reason why the law of falling bodies should be what it is. The early modern apostles of the Understanding expressed this by saying that the ‘cause of gravity is occult,’ and insisting that we must ‘feign no hypotheses’ about these supersensible questions. But when we talk about gravity as a real force we are already feigning a hypothesis unless we take the theory (with all of its difficult mathematics) to be just a shorthand description of a totality of ‘appearance’ from which predictions can be made deductively (and retrodictions too, of course). The theory is just a duplication of the intellect of the real structure of the world-order. The construction of this theoretical duplicate is called ‘explanation.’ So explanation is really tautologous. Physical reality and physical necessity are simply identified with the mathematical description and its resulting logical necessities.”

LostInParadise's avatar

The question can be broken down into three questions.
1. Why does mathematics apply so well in describing the world?
2. How did we come to understand the relationship between math and the world?
3. How did we create formal systems of axioms and proofs?

1. Nobody has a clue. Here is a great essay by the physicist Eugene WIgner on the Unreasonable Effectiveness of Mathematics

2. We are born with certain mathematical intuitions, as are some other animals. We intuitively understand the concepts of one, two and three, and have intuitions of ideas like “and” and “or”. We can credit evolution for this basic understanding.

The great civilizations of the ancient world developed the general concept of number. We are fortunate that the Babylonians left a permanent record of their calculations because of their use of clay tablets. Numbers started out as tally marks. They would keep records of transactions using a symbol of a cow for each cow or a symbol of a wheat stalk for a certain amount of wheat. These were originally clay figurines that were placed in boxes. Later this turned into the use of marks on clay tablets. Still later they got the idea of using universal symbols for numbers.

3. Formal mathematics happened in one place, ancient Greece. Nobody knows for sure who came up with the notion of proof. It was an act of extraordinary genius. We do know who came up with the idea of using axioms. That was Euclid.

gorillapaws's avatar

I think another way of phrasing the question is: “What is the epistemic justification for the axioms of logic and mathematics (i.e. the Laws of Thought)?”

On one hand the answer is that they are self-evident. On the other I think it can be argued that there is a kind of inductive logic at play. That if it were possible given A = B, B = C for A to not equal C somehow, then we would have run across a case where this would have happened. Given that this has never occurred in the entire history of human thinking, we can inductively conclude that this isn’t possible or is self-evident.

gasman's avatar

A crucial question, on which great minds are divided, is whether mathematics is discovered or invented. “God gave us the integers; the rest is the work of Man”—19C mathematician Leopold Kronecker is one point of view. But physicists have long remarked at how well mathematics describes physical reality, and even seemingly abstract pure mathematics sometimes turns out, long after its invention, to be applicable to physics in the real world.

Choosing axioms/postulates is not so easy. Even Euclid’s 3000-year-old parallel postulate (given a line & point not on the line, 1 & only 1 parallel line through the point exists) has been controversial. It can’t be deduced from the others so it must be assumed, in order to prove certain theorems about plane figures, but non-Euclidean geometries (where the postulate is false) successfully describe other real objects.

Euclid’s notion that “things equal to the same thing equal each other” is stated, in modern terms, as the transitive property of the equality relation: If a=b and b=c then a=c. Such logical underpinnings of math seem obvious as universal rules of the game. I don’t think you can remove the very basis of logical inference & still call it mathematics or logic.

If your hope is to settle on a simple system of axioms, from which all mathematical theorems may be deduced, your world was shattered in 1931 by Kurt Godel, who showed that there are always statements which can be proven neither true nor false; they are undecidable. This was a game-changer in terms of how we view logical systems that include numbers, especially the status of unproven mathematical conjectures. Self-consistency and completeness don’t necessarily co-exist. A good explanation is in Godel, Escher Bach by Douglas Hofstadter.

mattbrowne's avatar

Axioms are the creation of human beings and they can be useful if they help us explain natural phenomena.

gorillapaws's avatar

@mattbrowne so the law of non-contradiction (just one example) wouldn’t exist or apply if there were no sentient minds to think them? Surely they work more like gravity and apply whether we’re around or not.

LostInParadise's avatar

It is just so odd how life keeps imitating math.

mattbrowne's avatar

@gorillapaws – I know it’s an old philosophical discussion. Math is either discovered (like gravity) or invented (like a spaceship).

gorillapaws's avatar

@mattbrowne it reminds me of the metaphysical ontology of properties: does the number 43267423745736824836524474585698 exist only while a mind actively thinks it and then no longer exists when the duration of that that thought expires? This is a challenge the Physicalist has difficulty explaining.

wildpotato's avatar

@gorillapaws Psst: “metaphysical ontology” is redundant

gorillapaws's avatar

@wildpotato haha, good point!

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