To do mathematics, one obviously needs definitions; but, do we always need axioms?
I was thinking that a statement like
For all prime numbers, there exists a strictly greater prime number.
cannot be demonstrated computationally, because we'd need to check infinitely many cases. Thus, it can only be proven by starting with some axioms.
On the other hand, the statement
3 is prime.
could perhaps be proven without axioms, because there's only finitely many cases to check.
Thoughts, anyone?
I don't think you can do any mathematics at all without axioms, or principles which are equivalent to axioms.
Consider the statement "3 is prime". There are two definitions embedded here:
3 is the successor of the successor of the successor of zero.
a positive integer is prime if it has exactly two divisors.
The notion of the 'successor' is usually supplied by some axiom scheme such as Peano Arithmetic, in order to assert that there exists the integer zero, and a successor to any integer. Even if we set that aside, how do you propose to obtain an exhaustive search of all of the integers for possible divisors of 3, to ensure that there are exactly two such divisors? In practise, we reduce the search space to the integers 1, 2, and 3, because we may prove that multiplication has certain properties of monotonicity with respect to the total order defined on the integers by extending a < a+1 transitively. This is as much axiomatic as the sort of reasoning that allows us to infer that any finite list of primes is incomplete.
Conversely, because we may reason axiomatically to obtain a procedure of bounded length to find the divisors of any positive integer, we can also reason axiomatically to obtain an algorithm which is guaranteed to halt (and with a bound on its running time) for any list of primes a further prime which is not contained in the list. The fact that we cannot demonstrate by exhaustion that every prime is smaller than some prime afterward, is not any more problematic than the fact that we cannot demonstrate by exhaustion that any of the integers larger than three can be a divisor; they are the same sort of statement.
Generally, the only reason why any sort of "numerical experiment" could be used to ascertain any facts at all beyond mere existence theorems is because we adopt principles, either axiomatic or intuitive (that is, pre-axiomatic), which allow us to infer certain consequences from an investigation of only a finite search space. For instance, by investigation of a finite search space, we may show that 3 has at least two factors; but to show that it has no more is a universal statement, not an existential one, and so requires axioms which allow us to infer universal statements. And without principles which assert the existence of certain objects, even existence theorems cannot be proven.
Not enough reputation for comment so I'll give an answer:
The fact that a number is prime or not depends of the ring (i.e. axioms) to which is considered to belong. For instance 3 is prime if seen as an integer, but not if seen as a rational number (it is invertible). Also 5 is prime as an integer, but not as a Gaussian integer (of the form a+b*i, with a,b integers).
The bottom line is that the axioms are there, even if they are not mentioned explicitly.
Mathematics goes through periods of accumulation of facts of interests and axiomitisations which orders and organises these facts. Although traditionally in European curricula mathematics is held to have begun properly with Euclids axiomitisation of Plane Geometry, this would not have been possible without the prior accumulation of facts about Plane Geometry, or even that figures in the plane are of interest.
In modern curricula far more emphasis is placed on axiomatisations than on accumulation to the point where mathematics is held to be synonymous with axiomatisation.
A more contemporary example is the Italian school of Algebraic Geometry which was seen as intuitive; of course to the layman their work would be seen as abstract & technical. Several waves of axiomitisations followed of which the most important was the work by Zariski and by Grothendieck.
Another example is Category Theory. Many of the ideas in this school were already implicit in the work of many mathematicians before Maclane & Eilenberg made them explicit.
In your own example a certain philosophy has to be established first. There is a school of thought that allows only facts that are demonstrable - they are called the constructivists. In particular they disallow proofs by contradiction. For example, if one asserts that primes are finite in number and from that derive a contradiction - this proof would be disallowed. One needs to directly prove that there are an infinite number of primes.
A subsect of this thought are the ultrafinitists who argue that numbers above a certain threshold, perhaps not exactly specifiable are not constructible, and therefore do not exist. In this school the number of primes are actually finite in number, but the actual number is perhaps not precisely specifiable. One could say that in the foundational thinking, as per the ZFC axioms, they deny the axiom of infinity.
Yes, mathematics always needs Axioms. That is what mathematics is all about: Axioms and deductions.
Eliezer Yudkowsky has a nice dialogue on the topic here (it is part of a slightly longer article series available here)
The difference between Axioms and Theorems is that Axioms require exactly zero deductions to be performed.
There is a flip side: Axioms and Deduction Rules are always complemented by Models. Models are things that obey some set of Axioms: A bowl of pebbles obey Peano Arithmetic (albeit with an axiom of upper bound), so you call it a model of Peano Arithmetic.
Mathematics does not need axioms. Axioms have been invented only in order to consolidate and formalize the knowledge abstracted from the observation of reality, first in geometry, much later in arithmetic and other branches. But without any axioms every triangle we can draw has angular sum 180° and 1 + 1 = 2 - whether we note the result as 2 or II or 10 is irrelevant.
The describing character of axioms and the fact that axioms have to obey (and not to rule) mathematics becomes obvious from statements like: "And whether, in particular, Zermelo's axiom is true or false is a question which, while more fundamental matters are in doubt, is very likely to remain unanswered." (B. Russell) "Part of our aversion to using the axiom of choice stems from our view that it is probably not 'true'." (Peter G. Doyle, John Horton Conway)
Unfortunately meanwhile the pendulum has swung into an utmost insane direction where every axiom has to be tolerated as equally valid unless it contradicts other axioms.
If you were to hold that Mathematics cannot exist or be studied, or practiced--use your favorite verb for the context--without axioms, then nobody would have studied Mathematics until the Greeks starting not long before Euclid. I take it that proposition is sufficiently unpalatable to convince most people that Math can be studied without axioms. Any child knows, without the help of axioms, that 1+1=2 and does not understand this to be a result of earlier or more primitive truths like the meaning of 2 or 1+0=1.
Perhaps trivially one can call 1+1=2 its own ad hoc axiom, so that 1+1=2 is some sort of irreducible truth to anyone who hasn't yet learned a more standard collection arithmetic axioms. But then it's hard to see how this maneuver is substantially different from considering, for instance, "Either my sister went to the store or to a friends, but she's not with her friends, so she must have gone to the store," as its own axiom in a logical system--and that is also fairly obviously not right.
Notice also that the differential and integral analysis offered by Newton and Leibniz were developed with the notion of limits and infinitesimals, and these were used somewhat freely for a long time before a more rigorous foundation, the subject of topology, was formulated. At that point there still did not exist a completely formalized foundation for Algebra or sets, so if you think Math does not exist without axioms then we still hadn't discovered Algebra by the 16th century or even much later.
What an axiomatization provides is not substance but organization of ideas. It makes clear what your assumptions are and how you might trace the logical lineage of any claim.
To deduce proven statements in mathematics, you start with a list of axioms which are assumed to be proven by themselves, and then you use rules that let you create a new proven statement from other previously proven statements.
To have mathematics without axioms, you start initially with zero "proven" statements. Then you either admit that you have zero proven statements, which is not satisfactory, or you have rules that require zero previously proven statements to deduce a new proven statement. But then there is no real difference between having a rule that lets you decude a statement S from nothing, and an axiom that states S. So apart from a sleight of hand that hides axioms as "rules for creating proven statements without any previously proven statements", you need axioms.
This article, which is being expanded, shows clearly that axioms are not needed in order to create effective mathematics:
https://www.futurebeacon.com/rational.htm
Jim Adrian
You cannot always separate axioms from definitions.
E.g., the 3 properties of a distance metric (positivity, symmetry, and the triangle inequality) can be viewed as parts of its definition or as axioms.
The whole world is taught that there is no math without axioms. I am appalled.
Why don't people try it?
I suggest trying to define these terms first:
number, procedure, element, collection, set, ordered set, one, two, zero, numeral, and one-to-one correspondence.
You need the starting point of realizing that our perceptions include perceptions that are quantitative in nature. The undefined terms can be the starting point for formal definitions.
You can tell the difference between a single thing and a pair of things as surely as you can tell night from day or red from blue. Children reason and communicate with such perceptions as soon as they learn which words other people use to refer to them. We all have the ability to perceive a single thing, a pair of things, and many things. These terms are given meaning through experience in the world. Terms this basic cannot be defined by more basic terms. The terms any, some, more than, less than, at least, no more than, at most, and few refer to perceptions, not to formal definitions. These and related meanings are the starting point for mathematical definitions. These terms refer to empirical observations.
Jim Adrian
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