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Suppose some agent doesn't believe in the existence of the positive square root of 2.

Question: Is the agent's disbelief (or lack of belief) in the existence of +sqrt(2) irrational?

The question can be generalized to abstract objects of any kind, but this concrete case should suffice.

  • What do you mean by exist? By what definition does 2 exist, let alone sqrt(2)? – user4894 Mar 3 '14 at 17:41
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    You might want to specify your criterion of abnormality for beliefs. Otherwise, that makes no sense. A charitable reading is: is it irrational to not believe in the existence of irrational numbers? This might have been a slightly more useful question. – Hunan Rostomyan Mar 3 '14 at 17:42
  • the 2 is the real number 2 in the real number system of mathematics, sqrt(2) is also a real number found by doing operation sqrt() on real "2". Operation is unary, operation here is a function from the set of reals to itself. And set and function are from maths. – ReekMaths Mar 3 '14 at 17:54
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    @user4894 Existence isn't the issue here, it's the claim that believing/not believing in certain existence claims is 'abnormal'. You can see this by amending the supposed question as follows: "One who does not believe the existence of sqrt(2) [in ZFC], is abnormal." Still unintelligible due to 'abnormal'. While I think everything you've said is correct, I think your questions have misdirected the OP (evidence: the last revision of the question). So I'd suggest focusing on the fact that the 'abnormal' is the issue here, for a start. – Hunan Rostomyan Mar 3 '14 at 20:05
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    I've changed the question. I hope it's not too far from what the OP had in mind. Please consider retracting your votes if you feel that there is an answerable question there. – Hunan Rostomyan Mar 4 '14 at 8:03
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The question you asked suffers from two obvious problems.

  • i. missing criteria of abnormality for beliefs (please specify).
  • ii. missing explanation for the claim that most people intuitively 'feel' numbers.

I'll attempt to answer the question you should have asked instead, which starts with these 3 points:

(1) The principal square root of 2 exists.

(2) Agent α does not believe in (1).

(3) Agent α's lack of belief in (1) is irrational.

Your main question I took to be the following:

Question 1. Does (3) follow from (1–2)?

The answer, of course, depends on the logical structure of (1–2), so here is one way of analyzing it.


In (1), observe that the principal square root of 2 is an irrational number (for a proof, look at Example 1.1 in Baby Rudin, or in some other standard real analysis text). If irrational numbers are included in our universe of discourse, which is to say that we're in the universe of (at least) real numbers, then it will obviously be the case that the principal square root of 2 exists. That is because existential quantification will be over the domain of reals. But if we restrict ourselves to quantifiers that range over say rationals, then since rationals while dense nonetheless contain 'gaps', (1) will turn out to be false. At this point agent α can have a disagreement with us, depending on her answer to the following question:

Question 2. What's the universe of discourse? I.e., what does 'exists' range over in (1)?

As User4894 pointed out in the comments, if you understand by existence physical existence, which is to say that 'exists' ranges over stuff (at some level of interest) in the physical universe, then (1) will obviously be false, because irrational numbers aren't the sort of things which can be found in nature. Needless to say, it won't help to find a unit square (table say) and then say that its diagonal is the principal square root of 2. It won't even help to identify the number with the class (which being a class of physical objects is small enough to be a set) of diagonals of all physical unit square objects, because these objects are approximations of actual unit squares and nothing more.

In sum, if 'exists' in (1) ranges over any proper subdomain of reals, including ∅ in the physical scenario considered above, then (1) along with all the other irrational numbers can be said to not exist, so:

Observation 3. Premise (1) is true iff 'exists' ranges over domains not smaller than reals.

If we thus restrict existence, then (1) will be false. And then agent α's disbelief reported in premise (2) will be not only not irrational, but quite justified. But let us suppose the opposite, namely that:

Assumption 4. 'Exists' ranges over domains not smaller than reals.

Now suppose agent α doesn't believe in that, i.e., that (2) is true. Agent's disbelief in (2) would indicate that she understands something different by one of the constituent expressions in (1). For example, she might not understand how definite descriptions work, or how the square root function is defined, or the role the adjective 'principal' is playing, or the denotation of '2', or the meaning of 'exists'. If any of those conditions are present, then (1) won't have meaning to the agent, and she will naturally have no reason to believe it. So let's record this possible point of disagreement:

Question 5a. Does agent α understand the meaning of claim (1)?

A positive answer to that question is a start, but more needs to be settled. Because even if the agent understands (1), it might still be the case that she understands something quite different from what the rest of us understand. In that case we would hesitate to cast a judgement on α's disbelief in (1) unless we took into consideration the meaning she associates with (1). We want to further ask whether the meaning α assigns to (1) coincides with the meaning we assign to (1). So let's add that to our list:

Question 5b. Does the meaning α assigns to (1) coincide with the meaning we assign to (1)?

Suppose α understands the meaning of (1) and that meaning is the one we also associate with (1). I'll be assuming that (2) means that α believes that (1) is false, not that she has no belief in its truth. We take (1) to be true, we understand what α understands by it, but somehow she beliefs (1) to be false. Is this an irrational behavior, i.e., is (3) true?

At first, the answer seems to be 'yes'. After all, we understand the same things by (1), so either she doesn't understand how the criteria for the existence of the specified object are met or she does understand, but fails to believe in it. So, possible point of disagreement is:

Question 5c. Does α understand how the truth-conditions of (1) are met?

If the agent does not understand that, there might be some reason for it. Perhaps she's not capable of following logical inferences. (There might be physiological reasons for that.) If that's the case, then we wouldn't want to call α 'irrational', but as one who is simply incapable of seeing, for natural reasons, that the truth-conditions for (1) are satisfied.

The other horn of the disjunction is that while she understands the truth of (1) she somehow fails to believe in that truth. Here, at last, I think we've come to the crucial part of the question. We want to say that if (Questions 5a–c) are answered affirmatively, then the agent's disbelief in (1) is irrational. After all, we think (1) is true, α understands what (1) says (by the answer to Question 5a), α understands what we understand by (1) (by the answer to Question 5b), α understands that (1) is true (by the answer to Question 5c), and yet doesn't believe that (1) is true. Is there not some problem here? In other words:

Question 6. Must an agent believe every proposition she understands to be true?

Perhaps. The least we'd want to claim is the following, which we could call a 'principle of rationality':

Principle 7. If agent α holds proposition φ to be true, then α must not hold the belief that φ is false.

Earlier I interpreted (2) as 'agent α believes that (1) is false'. If that's the correct interpretation then we would want to say that the agent is indeed being irrational. But if the agent simply lacks the belief in the truth of (1), the answer depends on your answer to (Question 6). I think it's a reasonable principle of rationality, but it doesn't strike me as too obvious, so I'll let others talk about that.

† Thousand apologies for the length, possible typos and absurdities. Will certainly revise, shorten, and clarify asap.

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The mathematically interesting question here is when we need to use tools like numerical approximations, differential equations, measures etc. as tools specifically in Real Analysis in our general life.

I imagine 90% of people will never need to reuse what calculus they studied in higher education, never mind analysis in broad generality. So no. Finite arithmetic, and the natural numbers, is probably enough to believe in.

Of course, if you're a mathematician working in Set Theory, or involved in any kind of mathematically sophisticated science, then yes you should believe in irrational numbers, because of Cantor's proof of the existence of uncountable sets.

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There are mathematicians who accept only those entities as numbers which can be expressed without any uncertainty by finite digit sequences in some basis, i.e., rational numbers. That position is not irrational but contrary to common use.

In order to answer the question, first we have to determine what existence means. There are two alternatives with respect to numbers:

E1) A number exists if it can be individualized in mathematical discourse such that its numerical value can be calculated in principle by every mathematician without any error.

E2) A number exists if it can be individualized in mathematical discourse such that its numerical value can be calculated in principle by every mathematician with error less than any given eps > 0.

If you agree to E2, then sqrt2 exists.

If you accept the following grades of definition, then the sqrt2 is a well-defined number by D3. But people who require D2, don't accept sqrt2 as a number. These people opine that "number" is a patent of nobility that cannot be issued to values that cannot be represented by decimal systems. Of course however they accept that irrationals are existing as important mathematical objects.

D1) Some real numbers are extremely well defined: The small natural numbers like 3 or 5 can be grasped at first glance, even in unary representation.

D2) A real number is very well defined, if its value (compared with the unit) can be determined without any error, like all rational numbers the representations of which have complexity that can be handled by humans or computers.

D3) A real number is well defined, if its value can be determined in principle with an error as small as desired, i.e., the number can in principle be put in trichotomy with every very well defined rational number. The irrational numbers with definitions that can be handled by humans or computers belong to this class.

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