While there are some people (a part, if not all, of so-called finitists) who claim that no irrational number exists, it is not about the reality.
Topics
About: truth
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Does No Irrational Number Exist?
- 2: Why Irrational Numbers Are Required
- 3: The Shortcoming of "Expression-Oriented Mathematics", at Least Finite-Digital-Expression-Oriented One
- 4: The Limitation of Digitization
- 5: What Is an Approximation Without the Exact Value?
- 6: The Objection Against Irrational Numbers Seems Aesthetic
Starting Context
- The reader knows the background of this site.
Target Context
- The reader will understand what the fact that some amounts cannot be expressed as rational numbers means.
Orientation
There is an article on the reality, observations, and interpretations of them.
There is an article on the necessity for stating the major premise explicitly and honestly.
Main Body
Stage DirectionHere is Special-Student-7 in a room in an old rather isolated house surrounded by some mountains in Japan.
1: Does No Irrational Number Exist?
Special-Student-7-Hypothesizer
While we talked about a YouTube channel, "Insights into Mathematics", in an article (we said that the channel was very thought-provoking, being sometimes really educative, sometimes impressively beautiful, and sometimes absolutely unacceptable), the channel claims that no irrational number exists.
Special-Student-7-Rebutter
While the channel is sometimes very lucid, as it begins to argue against irrational numbers or some other certain things, it suddenly becomes so muddy that I cannot discern the logical structures of the arguments.
Special-Student-7-Hypothesizer
That is because the channel does not disclose the major premises.
The channel is like "As humans don't know well about big numbers, the natural numbers set doesn't exist!". Huh? There must be a major premise missing, like "Whatever humans don't know don't exist!".
The channel's objections against 'Cauchy sequence' and 'Dedekind cut' are mainly based on its objection against infinite sets, which is based on such hidden major premises, which are not shared by us, so, the arguments do not logically add up for us.
Special-Student-7-Rebutter
I cannot discern any reason why irrational numbers as finite algorithms do not exist.
Special-Student-7-Hypothesizer
The claim that not all the irrational numbers can be expressed as finite algorithms cannot of course dispute the irrational numbers that can be expressed as finite algorithms, and the fact that an irrational number can be expressed in multiple algorithms cannot of course deny the irrational number, as the fact that \(\frac{1}{3}\) can be expressed in multiple forms like \(\frac{2}{6}, \frac{3}{9}, . . .\) cannot deny \(\frac{1}{3}\).
Special-Student-7-Rebutter
That claim itself is dubious, as it may be just that the algorithm for an irrational number is not known to humans yet.
Special-Student-7-Hypothesizer
The only reason cited in the channel seems to be that irrational numbers as finite algorithms are cumbersome for calculations, if that could be called a reason.
Special-Student-7-Rebutter
There seems to be a hidden major premise like "Whatever cumbersome for humans don't exist!".
Special-Student-7-Hypothesizer
The channel is claiming also that the ticker tape of the infinite decimal of any irrational number cannot be contained in the universe, but also the ticker tape of the infinite decimal of \(\frac{1}{3}\) cannot be contained in the universe.
In fact, I do not see any essential difference between \(\frac{1}{3}\) and \(\sqrt{2}\), because any digit of each of them is determined, with only the difference of hardships for knowing it.
Special-Student-7-Rebutter
Obviously, the ticker tape tale can apply only to irrational numbers as infinite decimals by infinite choices, for which the ticker tape is the sole source of the determination of each number.
Special-Student-7-Hypothesizer
The real issue should be whether the infinite decimal is uniquely determined, not whether a finite algorithm exists, right?
Special-Student-7-Rebutter
I think so; a finite algorithm is just a way but not necessarily the only way of determining the infinite decimal.
Special-Student-7-Hypothesizer
I kind of agree that irrational numbers as infinite decimals by infinite choices are not valid, because such an infinite decimal is not determined until someone has finished choosing all the digits, which has not been done yet. The professor is talking about the ticker tape not able to be contained in the universe, but that is irrelevant; what is relevant is that nobody has finished choosing all the digits.
But an infinite decimal does not need to be determined by a human choosing all the digits digit by digit; obviously, fixing a finite algorithm determines the infinite decimal, but also fixing any more generally appropriate specification determines the infinite decimal.
Special-Student-7-Rebutter
When you say "more generally appropriate specification" . . .
Special-Student-7-Hypothesizer
For example, 'the diagonal length of unit square' is fine.
Special-Student-7-Rebutter
You mean, even if any algorithm is not presented, nevertheless, the amount is determined, which is what matters.
Special-Student-7-Hypothesizer
Yes. And also 'the height of this penguin' is fine.
Special-Student-7-Rebutter
Ah, it is not the matter of whether humans do or can know the exact height, you mean?
Special-Student-7-Hypothesizer
Yes. Whether humans do or can know the exact height does not influence the existence of the exact height in the reality, which is what matters. In fact, the penguin does not become taller or shorter just because I have mis-measured the height, or the penguin is not in a mysterious state of having no height just because I have neglected to measure the height.
Special-Student-7-Rebutter
That is the point we really disagree with the professor: he is like "Something doesn't exist just because humans don't know it!".
Special-Student-7-Hypothesizer
And each point on a coordinates axis determines the infinite decimal by virtue of existing there.
Special-Student-7-Rebutter
You mean, we, humans, do not need to pick up a point in order for the point to exist.
Special-Student-7-Hypothesizer
Of course: The point does not begin to exist as we have picked up the point, but as the point has existed, we can pick it up.
Special-Student-7-Rebutter
So, any irrational number is defined all right by a point on the axis.
Special-Student-7-Hypothesizer
It will be needless to say that it is not the matter of whether humans can know the exact coordinate of the point.
And if you begin to claim like "Humans can't draw any exact coordinate axis." or "Humans can't pick up any exact coordinate point.", I remind you that it is not about irrational numbers any more: you cannot do so for rational coordinates either.
2: Why Irrational Numbers Are Required
Special-Student-7-Hypothesizer
To answer to "Why irrational numbers are required?", for example, otherwise, the diagonal length of unit square would not exist.
Special-Student-7-Rebutter
The professor seems to be promoting using "quadrance" instead ('Math Foundations 133: Higher dimensions and the roles of length, area, and volume'), meaning that we should talk in terms of squares of length, instead of in terms of lengths.
Special-Student-7-Hypothesizer
But, using "quadrance" does not solve the problem. For example, extend the 2 "quadrance" diagonal with the 1 "quadrance" line segment, and the result line segment will be a \((\sqrt{2} + 1)^2\) "quadrance" line segment, with the irrational "quadrance". So, "quadrance" can be irrational anyway.
Special-Student-7-Rebutter
By the professor's mathematics, you cannot draw the 1 "quadrance" line segment in the diagonal direction, because he does not admit irrational numbers plane but rational numbers plane, in which the 1 "quadrance" line segment cannot exist in the diagonal direction.
Special-Student-7-Hypothesizer
Being told "you cannot", why can I not turn 45 degrees and draw the 1 "quadrance" line segment in the now horizontal direction?
Special-Student-7-Rebutter
You seem to have adopted a new coordinates system with the new horizontal direction as an axis.
Special-Student-7-Hypothesizer
Of course.
Special-Student-7-Rebutter
According to the professor's mathematics, you cannot adopt the coordinates system by which the diagonal's "quadrance" keeps being '2'.
Special-Student-7-Hypothesizer
Huh? What do you mean by "you cannot"? I can adopt any arbitrary coordinates system, right?
Special-Student-7-Rebutter
I think you could, but the square would not appear on the rational numbers coordinates system, as its vertices would have irrational coordinates.
Special-Student-7-Hypothesizer
"would not appear"? Where has it gone? Has it disappeared from the reality?
Special-Student-7-Rebutter
As the square is not moved or changed in any way, it cannot have disappeared from the reality if it had ever existed before you had adopted the coordinates system, but it does not appear on the coordinates system.
Special-Student-7-Hypothesizer
So, the coordinates system cannot capture a part of the reality.
Special-Student-7-Rebutter
In fact, most of the reality slips through the net of the coordinates system, as most points have irrational coordinates.
Special-Student-7-Hypothesizer
Is that OK?
Special-Student-7-Rebutter
The professor is saying that you cannot help but accept that the mathematics is so.
Special-Student-7-Hypothesizer
Let him speak only for HIS mathematics; the standard mathematics is not so.
Obviously, it is not the matter of whether we should use length or "quadrance"; it is the matter of that HIS mathematics cannot capture most of the reality.
Special-Student-7-Rebutter
It is that irrational numbers are required in order to fully (or at least better) capture the reality.
3: The Shortcoming of "Expression-Oriented Mathematics", at Least Finite-Digital-Expression-Oriented One
Special-Student-7-Hypothesizer
While the professor is promoting "expression-oriented mathematics" over traditional "object-oriented mathematics" (Math Foundations 77: Object-oriented versus expression-oriented mathematics ), its obvious shortcoming is that not every object in the reality can be expressed finitely in digital form, so, some objects in the reality slip through the net of the "expression-oriented mathematics".
Special-Student-7-Rebutter
In fact, he is promoting "finite-digital-expression-oriented mathematics", which is the point.
For example, any finite line segment itself, for example a 2 "quadrance" horizontal line segment, is finite, contained in a finite space region, and nothing is wrong with the line segment, whether it can be expressed in a finite digital form or not.
As an object has no obligation to be expressed in a finite digital form, the object can slip through the net of that mathematics.
Special-Student-7-Hypothesizer
"finite digital form" is just a matter of human convenience, which the reality does not care.
If someone is fixated on finite digital form, he or she could tend to declare that something did not exist just because it could not be expressed in a finite digital form.
Special-Student-7-Rebutter
The wording is bad: he or she should not say "Something does not exist" but should humbly say "HIS or HER mathematics cannot see something.".
4: The Limitation of Digitization
Special-Student-7-Hypothesizer
Digitization is a way of establishing a correspondence between an amount and a number.
Amount is for example a length, and any length does not have any obligation to be digitized, but humans want to digitize the length for the sake of their convenience.
Digitization depends on the arbitrary choice of unit, and a unit square is so just because someone has chosen the unit to make it so, and the diagonal length inevitably corresponds to \(\sqrt{2}\), an irrational number, because of the arbitrary choice; if that someone has changed his or her mind and chosen the diagonal length as the unit instead, now the diagonal length corresponds to 1, a rational number, and the side length inevitably corresponds to an irrational number.
If you are unhappy with irrational numbers, nothing is wrong with any length or any line segment, but the digitization method brings in irrational numbers.
The reason why the set of all the rational numbers does not cover the set of all the decimals is that any rational number is inevitably a finite decimal or a circulating decimal, but obviously there are non-circulating infinite decimals, for example '0.101001000100001...'.
The reason why any rational number is inevitably a finite decimal or a circulating decimal can be understood if you think of how you compute the decimal from the fraction: the residue of the calculation for the previous digit is used for the next digit calculation, but the residue can be only from 0 to the denominator minus 1, so, the residue cannot help but eventually become 0 or return to the one for a previous digit calculation, and the further calculations become cyclic.
Special-Student-7-Rebutter
In fact, as far as human convenience is concerned, any infinite decimal is not so much different in inconvenience, whether it is an irrational number or a rational number: we have to endure with an approximation any way.
Special-Student-7-Hypothesizer
Yes. While the professor attacks irrational numbers, I do not see any essential difference between '0.333333333333333...' and '0.101001000100001...' and between '1/3 + 1/5 = 8/15' and '\(\sqrt{2} + \pi = \sqrt{2} + \pi\)', as '\(\sqrt{2} + \pi\)' is a finite expression if '8/15' is so.
The limitation of digitization is that the decimal can be infinite, whether it is in a circulating way or not.
Special-Student-7-Rebutter
"limitation" is just with respect to human convenience, as nothing seems to be wrong with any infinite decimal, logically speaking.
5: What Is an Approximation Without the Exact Value?
Special-Student-7-Hypothesizer
The channel is claiming like "An irrational number is OK as an approximation, but not so as an exact value.", which is utterly unintelligible for me.
Special-Student-7-Rebutter
An approximation toward what? In the prevalent definition of approximation, any approximation is an approximation toward the exact value. If the exact value did not exist, what would the approximation be approaching to?
Special-Student-7-Hypothesizer
The usual logic is: if there is ever an approximation, there must be the exact value, as "approximation" is nonsense without the exact value, and the exact value is nothing but an irrational number.
I do not understand the logic of the professor a bit.
6: The Objection Against Irrational Numbers Seems Aesthetic
Special-Student-7-Hypothesizer
The professor's objection against irrational numbers does not add up logically.
After all, the objection seems aesthetic; probably, irrational numbers are offensive to his aesthetic.
I have admitted that his mathematics is beautiful, at least in a way.
But ugly things do not vanish from existence just because they are ugly.
The professor talked about inconvenient truths about \(\sqrt{2}\) in a video ('Math Foundations 80: Inconvenient truths about sqrt (2)'), but the people who accept irrational numbers are not dismissing inconvenient truths, while the professor is saying that we should dismiss \(\sqrt{2}\) just because it entails inconvenient truths, so, is ugly.
