In math, numbers and addition are logically defined by Zermelo Set Theory, a small group of axioms upon which everything else can be built. Could it be possible to have a working theory, (in any field not just math), without any preexisting axioms?
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It is hard to imagine that: every theory must have some basic (undefined) concepts with which define new ones and some basic statements (axioms) regarding the previous concepts that implicitly define "how to use" the basic concepts. See e.g. Spinoza's Ethica, ordine geometrico demonstrata.– Mauro ALLEGRANZACommented Jul 5, 2018 at 19:44
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I think yes, right as language (that can have no definitions - context-sensitive), but this theory would be unformalizable.– rus9384Commented Jul 5, 2018 at 20:02
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1Zermelo axioms were not even formulated until 1905, mathematics existed long before that and much of it was not axiomatic at all. Much of biology is not likely to be mathematizable or axiomatizable in principle. So the answer is a trivial yes. Unless by "preexisting axioms" you mean some preexisting conceptual background in some vague sense, in which case the answer is trivially no. Is that really what you are asking?– ConifoldCommented Jul 5, 2018 at 20:56
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In logic and mathematics anyway, no formal theory can exist without axioms or rules of syntax and inference. Otherwise, how could you even write a single line of proof? Somehow you need to justify even that first line. You could not even write $0 \in \N$. You have no rules of syntax to verify whether this is even a valid formula. And you have no means to justify writing it as a line of proof.– Dan ChristensenCommented Aug 10, 2018 at 2:50
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1Such theory, I presume, would act as an axiom itself.– Yechiam WeissCommented Nov 5, 2018 at 6:45
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5 Answers
This entire question is a case of classical equivocation.
Theories in sciences don't have axioms. In fact theory, in the sense of Set Theory, or The Theory of Complex Variables has nothing in common with the notion of a scientific theory whatsoever.
A scientific theory is a proposed explanation for a set of phenomena, and a theory in the mathematical sense does not explain anything, it is a subject in itself.
In math, a theory is a major part of a field of study that is somewhat separable from related parts of the same field. It is often defined as the body that proceeds from the understanding of a given model.
The modern way of looking at a model is as the embodiment of a set of axioms. But that is not the only approach to take. A lot of people do not think of Euclidean geometry as what proceeds from Euclid's axioms, but what proceeds from a given shared understanding that humans develop naturally by living in macroscopic three-dimensional space. So by that notion, things like geometry, real analysis and number theory do not really involve axioms. The axioms are created to solidify our understanding, but they are in some sense superfluous, and a concession to a level of excessive modern formalism in logic, and to the success of abstract algebra. These domains that focus intently on a single model really have that model in a more informal, but shared set of ideas.
To think of addition as defined by Set Theory ignores millennia of history during which people really did math.
So:
These two things don't really overlap.
Even in math, axioms are only an aspect of a given approach, although an extremely common one.
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If we have proofs in geometry (rather than just observations from living in a 3+1 spacetime that's locally very close to flat), we need something to prove from. Without that, what we've got is a science of measurements. We can't prove that the internal angles of a triangle sum to half a circle, we'd have to inscribe more and more precise triangles and measure more precisely to determine that (with very high probability) it's always between 179 and 181 degrees. Commented Nov 7, 2018 at 18:29
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1@DavidThornley You can adopt that perspective, in retrospect, but we evolved into having axioms from having working geometry. So it seems like you are either have to declare the first thousand years of the subject not to actually qualify as mathematics, or you have to admit the axioms are there just to make things more reliable, and are not strictly necessary. We can idealize our experience, but that does not make the idealization prior to the actual experience. And we can choose other kinds of idealization than axioms.– user9166Commented Nov 7, 2018 at 20:11
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@DavidThornley Consider the pictorial 'folding' proof of the Pythagorean theorem. (faculty.smcm.edu/sgoldstine/pythagoras.html) No axioms, only an implicit trust in our ability to handle idealized pictorial objects, ad an understanding of the notion of product as area. Is it not a proof? It was certainly considered a proof at the time it was current.– user9166Commented Nov 7, 2018 at 21:12
Is the answer to this question, to ask another question? Can a question exist in nothing?
ie. it takes an intelligence to ask a question and an intelligence exists on certain principles that pre-exist that intelligence, upon which it relies.
The foundation is an axiom. All we know is this very experience. One argument for evolution is it came out of nothing. But equally within the theory there are assumed rules and points of stability that exist for no known reason, and therefore are axioms. There is a faith or belief that if something exists, it exists in isolation and cannot have been created, even though its origin is unknown and what it is is also unknown, just its apparent effect that can be measured.
I suspect we like absolutes, and conclusions rather than processes and a process of becoming and discovering which never ends, though this is our experience of life.
Could it be possible to have a working theory, (in any field not just math), without any preexisting axioms?
I doubt it. Without some set of axioms, the observer will almost always draw a conclusion that is random in relation to the event observed; further, in the absence of guidance from fundamental principles, the observer can never understand when a conclusion is accurate, even when it is so.
“An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.” (Wikipedia, Axiom) I think of Euclid’s Elements as stating axioms of mathematics, particularly of geometry. Without them, a mathematician could not conclude, one way or the other, that the three angles of a triangle total 180 degrees.
I think of the Laws of Thought as axioms. Without them, there would be no assumptions about identity, non-contradiction, or the excluded middle. Such an absence of guidance would permit any number of random theories, none of whic could be attached to reality, nor to any larger theory about the world.
answered Nov 7, 2018 at 3:51
I don't think so, since a solid theory is a kind of intellectual construction and axioms are the pillars that support it.
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If you have any reference that would support the position that axioms are needed this would give the reader a place to go for more information and support your answer. Saying they are needed does answer the question, but it doesn't provide a justification for why they are needed which is what I think the OP is looking for. Commented Sep 1, 2019 at 22:55
Existence precedes essence. Axioms are great, but I put it to you that in practice they come after finding an interesting theory, in the process of generalising and universalising it.
If the laws of physics existed before the universe, where did they exist? Theynare the universe, and the universes behaviour is the laws. Theybare true in detqil and axiomatically at the same time, without hierarchy. Things just exist, and how they exist are the laws. We look for patterns, and what is invariant we call an axiom. But we begin with the pattern, generalise it a little, learn more, get deeper axioms, etc etc. The world continues to work without us knowing any axiom, and the world is the theory of all theories.
answered Oct 6, 2018 at 0:55
