November 2018
My apology for the completely messed up format! I have not figured out the best way to import text from Latex here :/
1 Introduction: Arguments of Wildberg and its Consequence
Recently, Australian mathematician N.J.Wildberg has made some unconventional claims on modern theoretical and pedagogical mathematics. He has gained a lot of influence through the internet, not only among mathematicians, but also among a lot of undergraduate math students. A lot of his videos ended with the message and encouragement towards math undergraduate students to reexamine and revise the fundamentals of mathematics.
In this paper, I will examine and argue against the most central claim of Dr.Wildberg, that a mathematical infinity does not exist. The recognition of an “infinity” is crucial for the study of a lot abstract mathematical objects and hence a doubt on infinity is full of consequences. For example, based on the assumed legitimacy of this central claim, Dr. Wildberg proceeded to reject the notion of the continuum, the only well-known example of a non-denumerable set, based on that the complete arithmetical notion of the continuum requires that one admits the legitimacy of an infinity of countably many successive arbitrary choices. It is imperative that we scrutinize the ideas of Dr. Wildberg.
2 Direct Refutation of Dr.Wildberg
In this section, I will be examining the crucial statements and examples Dr. Wildberg used in his denial of infinity. I will attempt to melt down his arguments by looking for logical inconsistency within themselves, and challenging his definitions.
2.1 Non-representation implies non-existence?
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In one of his lecture series, Dr. Wildberg wrote down the number z = 10(^10^(10^(…))) + 23 and made the point that even though ”z” carries limited amount of information (evidently since he is able to write it down), most numbers between 0 and ”z” are of such great complexity that they cannot be expressed in any fashion whatsoever in the known universe, so their existence is therefore doubted. He further made the observation that because of this computational difficulty, prime factorization is only true up to a certain point. For example, “z” cannot be factorized due to the limitation of computation.
I think the fault of this argument is that the limit of representation is not the limits of existence. But let us take a step back to examine the ground that we are having the argument on, which is that the number that we are writing down on paper or with a keyboard is itself a mathematical object—this ground is not solid. “Two” is not a mathematical object, neither is ”2”, neither is ”II”, these are names. The number two is an intrinsic property of the physical world. It marks the natural characteristics distinguishing two apples from three apples or one apple. Hence when Dr.Wildberg used the ”z” example, he had merely successfully demonstrated that the representation scheme of numbers has space of improvement.
The representation of any object has no logical link to either the existence or the properties of that object. We have seen the difference made by representation that can not be attributed to be the property of the computed. Take Presburger and Skolem arithmetic as an example. While multiplication is irrational with Presburger arithmetic, it is rational with Skolem. Hence the properties of computation of multiplication needs to be discussed within representational framework, but not within itself. It is neither rational or irrational within itself. Analogously, when Dr.Wildberg makes statement about prime factorization, he is merely describing our current 64 bits computation scheme, but not about prime fac-
torization itself.
From another angle, we can also prove that most irrational numbers that exist do not
have a representation with a counting argument: Since represent-able irrational numbers
such as sqrt(2) are encodable hence countable, and irrational numbers are uncountable, we can conclude that most irrational numbers do not have an representation.
If we recognizes the mathematical existence[1] of big numbers, every arguments follows naturally. Since we never proved prime factorization of any number with presenting prime factors of it at the first place, Dr.Wildberg’s claim on the illegitimacy of prime factoring big numbers is neither here nor there. This argument of limitation of computational power is at best a misunderstanding of the definition of computability, at worst an illogical argument.
2.2 Ball game paradox
In this paradox Dr. Wildberg gave, we have an infinite pile of balls in the corner of the room. We now take 10 balls at a time, throw one out and put nine in the box. At the end when we finish taking the balls, he stated, we will have infinite number of balls in the box. In the second scenario, we take ten balls, put all in the box, and pick one ball to throw out. After we finish, we will only have two balls in the box, because every time we throw out ball 3, ball 4, ball 5... etc. Hence we get a paradox.
It is not hard to see that the mystery of this paradox breaks when Dr.Wildberg used the word ”finish”. By attributes of infinity, we cannot The setting of ”an infinite pile” is inconsistent with the description of the procedure, namely that we can finish it within a finite time frame and contrast the number of balls in the two boxes. However, the initial setting of infinite number of balls does not deter us from a consistent and complete analysis. Define the time we take to complete one iteration of the actions to be one time frame. At time-frame T, there are T balls that we threw out, and 9T balls in the box. As T approaches infinity, T and 9T both approaches infinity. As far as we know, the two scenarios are
consistent.
2.3 A polynomial without solution
Dr. Wildberg has cast doubt on the Fundamental Theorem of Algebra by making an argument that the function x^5 − 2x + 3 = 0 does not have a real root. In the video, he also challenged anyone to give him the root, claiming that is the only way to falsify his argument.
The inconsistency of this argument is that Dr.Wildberg is trying to be a constructivist and a falsificationist at the same time. Due to this inconsistency, the simplest refutation to this argument is just a play of words. If it can be insisted that the absence of a solution is the proof of absence, which seems to be the ground logic that Dr.Wildberg employs, then we can simply apply Newton’s Method finitely many times (instead of infinitely, which he does not acknowledge), but enough times such that given the number of digits of x, computer can no longer computer x^5−2x +3 to compare it with 0. Then in his own term, due to the overwhelming complexity of this falsifying proof, there does not exist such a falsification. Hence x^5 −2x +3 = 0 has a root. We do not even need to use the complete notion of R to debate against Dr.Wildberg, due to the ubiquitous inconsistencies within his own arguments.
Additionally, in Dr.Wildberg’s paper ”Set Theory: Should You Believe?”, he wrote that ”The onus is on us to demonstrate that our notions make sense, instead of challenging someone else to find a contradiction”. The burden is hence on him to present constructively a proof of the non-existence of the roots. I therefore do not take this as a valid argument for the denial of the Fundamental Theorem of Algebra.
It is understandable that Dr.Wildberg does not have to make mathematically rigorous arguments in order to be achieve prevalence on the internet, and the nature of the content (Youtube episodes) almost impedes on the level of rigorousness one can expect to achieve. However, special caution do need to be taken, since a majority of Youtube consumers are college students, including a large number of math undergraduate. Misleading content as such makes more harmful impact than one would expect.
3 Direct Rejection of Ultra-finitism
Dr. Wildberg’s claims are neither sound nor novel. He is an ultra-finitist, and ultra-finitism has been under discussion for a long time. Ultra-finitism is a form of finitism that is even more extreme in that it denies those constructs whose construction cannot be physically performed (such as a number that no human could have time to calculate). Along the development of history of mathematics, a lot of famous mathmaticians including Gauss and Kronecher shared finitist thoughts. In this section, I will try to refute only ultra-finitism from a more modern standpoint.
3.1 Infinite Data Structure
To refute ultra-finitism, I hence first look at a computation scheme of infinity on modern computer. In Standard ML, a physical construction of the set of natural number has been realized. Consider the following data structure for the representation of natural numbers:
datatype 'a lazylist = Cons of 0a ∗ (unit → 'a lazylist)
Define the helper function ”nat”:
fun nats n () = Cons(n, nats(n +1))
Then the set of natural number can be represented as:
val Nats = Cons(0, nats 1)
If a simple ”N” on paper is considered to be too abstract, ”Nats” is an implementation of all natural numbers. This construction takes advantage of the ”call by value” characteristic of functional languages, and delay the evaluation of the function ”nats” with an argument type of unit. We hence represent this infinite set with finite data structure.
An ultra-finitist may still argue that we are not able to evaluate ”Nats” arbitrarily many number of times, hence ”Nats” still represent a finite set. But that will result in a pattern matching failure. Assume for the sake of contradiction that ”Nats” represent the finite list of [0,1,2...,x-1]. Consider the x-th time we invoke ”nats”, we are expecting it to return a list of the pattern
Cons(x −1,nil)
However, the empty list nil is not part of our data structure, we will hence fail to pattern match. We hence conclude that ”Nats” represent a non-finite set of numbers on our com-
puter, adhering to the ultra-finitist constructive doctrine.
Then, with the assumption of the dichotomy of ”finite” and ”infinite”[2], we conclude that ”Nats” is physically an infinite set on our physical computer.
3.2 Axiom of Infinity—the mathematical existence
My second response to ultra-finitism uses one of von Neumann–Bernays–Godel axioms,¨ the Axiom of Infinity. In the language of Zermelo-Fraenkel, the axiom is as follows:
∃I(∅∈ I∧∀x ∈ I((x ∪{x}) ∈ I))
In this construction, set I forms a bijection with the natural numbers if the successor of x is defined as x ∪{x}. This axiom asserts that there is a set I that contains 0 and is closed under the operation of taking the successor; that is, for each element of I, the successor of that element is also in I. If Ultra-finitism has truth in it, then the Axiom of Infinity will be negated, by the contrapositive of the statement that the Axiom of Infinity implies the construction of all natural numbers. But what argument does Ultra-finitism has against the Axiom of Infinity? A mathematical existence once induced cannot be denied unless it leads to contradiction. For the ultra-finitists who disagree with this statement, I ask what evidence they have that ”1” actually exists? Sure we can represent it as 0001, but I can also represent anything with 0001, that is not why everything exists mathematically. Hence the only way to reject this particular axiomatization is to present a logical contradiction in our current mathematical framework. Without presenting a contradiction, to reject the Axiom of Infinity is not to reject ”infinity”, but to reject axiomatization of any type.
4 Conclusion
In this paper, I have shown that the argument of N.J. Wildberg, that infinity does not exist, is not convincing. Further, I directly presented the representation of infinity in modern computation. All in all, I think it is imperative that we base mathematical applications on correct abstract theoretical mathematics, and not the other way around.
Using the example of representation of natural numbers in Standard ML, I have attempted to present a direct counterargument to the Ultra-finist argument that infinity has no physical existence in the physical world. And using the Axiom of Infinity, I have attempted to show that ultra-finitm has not only contradicted the mathematical existence of inifinity, but also the fundamental methods of mathematics. To accept ultra-finitism is to accept a foundation-less field of mathematics, where we are forbidden to define or axiomatize. I hence conclude that Dr.Wildberg has not presented any convincing argument, not only because of the inconsistencies within his own argument, but also because of his ultra-finitist assumptions.
Future research should be devoted on more thoroughly refuting finitism as a whole,
instead of just ultra-finitism.
Acknowledgement
I give my sincerest gratitude to Professor Klaus Sutner and instructor Chris Grossack. Without them I could not have made it through the semester and eventually finished this final project.
References
[1] Horsten, L. (2017, September 26). Philosophy of Mathematics. Retrieved from https://plato.stanford.edu/entries/philosophy-mathematics/Int
[2] Njwildberger. (2014, September 28). Infinity: Does it exist?? A debate with James Franklin and N J Wildberger. Retrieved from https://www.youtube.com/watch?v=WabHm1QWVCA
[3] Njwildberger. (2017, January 07). The fundamental dream of algebra — Abstract Algebra Math Foundations 216 — NJ Wildberger. Retrieved from https://www.youtube.com/watch?v=QSZsTeO-C1o
[4] Wildberger, N. J. (n.d.). Set Theory: Should You Believe?
[1] Hereby define mathematical existence to be not introducing a logical contradiction given a particular set of rules. [2] This dichotomy is embedded in the definition of infinity. It has nothing to do with the existence of an infinite set, which may be commented upon by constructivists, but is merely definition.
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