# What created the bias in humans that make us intuitively claim the refutation of the Continuum Hypothesis?

So I was watching this youtube video.

We've got a well ordering of the real numbers but just between zero and one that'll do okay now comes a little statistical argument. You and I are gonna throw darts we're gonna throw darts at the real line between zero and one. It's a thought experiment can't do it in reality but i thought it's better I throw a dart and I get a number p somewhere between zero and one you throw your dart and you get a number q also on the line between zero and one. Now I'm going to make the following argument your dart hit the number q which is going to be later in the well ordering than my dart p why because there's only a countable infinity at most of numbers earlier in the well ordering from p and an uncountable infinity after and so so that you you might be earlier than me but the probability is zero. Now you've hit number q and you can say exactly the same argument you're going to say the chance that I'm earlier in the well ordering is zero and the chance that it's later is is one now we've got an absurdity!

The assumption that there are aleph one real numbers if there were aleph two it wouldn't be a problem. They have landed in an uncountable there might be an uncountable number of points earlier than me and an uncountable earlier than you so we couldn't argue. We wouldn't be able to do that the trick is when you've got a countable infinity of numbers.

But the continuum Hypothesis states: There is no set whose cardinality is strictly between that of the integers and the real numbers.

He later on explains all the elements of the argument independently can be formalized to ZFC (such as randomness, statistical independence, etc). But the argument as a whole cannot be formalized. So it seems we are using intuition to arrive at the conclusion!

Now I assume the following:

1. My intuition was shaped by evolution.
2. The Continuum Hypothesis does not affect any physical outcome.
3. All humans would agree this intuitively this argument refutes Continuum Hypothesis.
4. Evolution is a physical process which is agnostic (works the same regardless) of the Continuum Hypothesis.

## Question

What created the bias in humans that make us intuitively claim the refutation of the Continuum Hypothesis?

• AH lemme fix that Commented Feb 18, 2022 at 13:32
• "Solomon Feferman has argued that CH is not a definite mathematical problem. He conjectures that CH is not definite according to this notion, and proposes that CH should, therefore, be considered not to have a truth value." Joel David Hamkins proposes a multiverse approach to set theory and argues that "the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and, as a result, it can no longer be settled in the manner formerly hoped for". Commented Feb 18, 2022 at 13:40
• Thus, IMO, to say "the bias in humans" regarding CH is not correct: humans (in general) have no idea (and no interest) in CH... Commented Feb 18, 2022 at 13:42
• Also mathoverflow.net/q/49721/64357 the author also seems to share the notion "On the other hand there is something intuitively true on Freiling's argument." Commented Feb 18, 2022 at 14:07
• I'm not saying that Freiling is right nor wrong... I'm saying that there is no "common sense" intuition about CH :-) The interesting post you have linked has an answer by the same Joel David Hamkins of my comment above: "My view is that any philosophical, pre-reflection or intuitive concept of probability will have a very fundamental problem..." Commented Feb 18, 2022 at 14:14

## Short Answer

Cognitive linguists would argue the bias of human mind which leads to biases in mathematical thinking is called the conceptual metaphor, particularly the Containment Metaphor. This might best be understood as the idea that the human mind is predisposed to think of abstractions as things in space instead of processes. Philosophically, to think the opposite is heterodoxy called process philosophy in the Western Philosophy because to give primacy to change over static structure is counterintuitive to the beginning thinker.

## Long Answer

Let's start with math history. Originally, Platonists argued that an Aleph is somehow real and reified, a static entity disembodied and disconnected from the human mind. Neo-platonic thinking affirms this. On the other hand, contemporary mathematical philosophers like Luitzen Brouwer (and constructivists like me) claim that an Aleph is nothing more than a process of comparison, a psychological extension of human thinking. I would advocate that the former way of thinking is a bias of the mind.

So unless one embraces constructivism fully, one is open to the bias that numbers are things when a more sophisticated view in mathematics is numbers are processes. That's why there was so much difficulty accepting incommensurable numbers, 0 or i, why it took so long to discover PA, and why many mathematicians continue to think about numbers as objects in a space. Analytical rigor in mathematics happened around the same time as psychologism in mathematical foundations, and I don't think that that is a coincidence. In fact, think about the metaphor you use. You fall back to conceptualizing an infinite set of points as a physical place (line drawn on a wall) where physical objects (darts) can be placed. This is the essence of the Containment Metaphor. Sets in set theory are built on this metaphor. (See Where Mathematics Comes From for more information on the four fundamental conceptual metaphors.)

'Why objects in a space?' then becomes the question. What is there that biases the human mind towards viewing things as objects in space, even when they have no physical embodiment whatsoever? George Lakoff and other cognitive linguists would argue that conceptual metaphors are inherent to cognition as a result of pressures from the evolution of the mind. What is this conceptual metaphor in more detail? It's just a mapping between conceptual primitive between domains. From WP:

A conceptual domain can be any mental organization of human experience. The regularity with which different languages employ the same metaphors, often perceptually based, has led to the hypothesis that the mapping between conceptual domains corresponds to neural mappings in the brain.1 This theory has gained wide attention, although some researchers question its empirical accuracy.

So, the simplified version to explain the bias might be like this. The human brain evolved to not get eaten on the African plains. To survive it evolved a quick version and a slow version (an idea reiterated in Thinking, Fast and Slow by Kahneman), and the fast version has a tradeoff between seeing 'things' quickly at the cost of see 'things' that aren't there. So, our mind sees the lion in the brush on occasion when there's none, because it's better to have a false positive and live, than a false negative and be eaten.

Now, what system predominates the human sensory experience? Stereoscopic vision. Unlike hearing and smelling, it can provide input to the mind at tremendous distances. And two planes of light that fall on the fovea overlap to create a 3D 'mental picture' of space. And so, the visual cortex is profoundly evolved in humans and central to cognition. So, is it so far fetched when language came along, that there's a bias to express utterances about ideas that are biased towards 'spatial' thinking involving 'objects' perhaps moving along 'trajectories'? Not at all if evolution builds the layers of language on top of the neurons that are already highly evolved.

By the time Plato comes around, he arrives at a fairly intuitional picture of numbers. They're objects like the objects they describe, and somehow our mind in this space is in touch with the Forms in some Realm which is another space somewhere else. It's rather similar to the idea that the human body is animated by a 'thing' or 'vital force', called a 'soul' that must also be an object that even survives the passage of the body in this space to another space after life.

Now, along comes the concept of empirical evidence. And vitalism survives as a scientific hypothesis until the 20th century. Of course, science rejects the 'soul' as non-empirical, an article of faith (and many scientists reject faith as a path to knowledge). But ask any Catholic priest if a soul is a real thing, and they'll answer, yes. Mathematicians are overwhelmingly platonic in their thinking. Linnebo's work Philosophy of Mathematics overwhelmingly supports this idea, that numbers are things to some extent. And that's why set theory is the preferred method of organizing mathematical thought. It's natural to think of number as things contained by space.

Now, does conceiving of numbers as collections of things as opposed to methods of processing data interfere with resolving the CH? I would argue yes. I would argue that the great mathematical revolutions have all occurred as extending the mathematical concepts beyond a number-as-things model. Lakoff and Nunez in their book Where Mathematics Comes From give a peek into the world of ideas like what does Euler's identity really mean when it relates Pi, i, 0, 1, and equality? And when they do it, they do process-oriented as opposed to trying to think as numbers-as-things.

Intuition is at the basis of all mathematics as noted by Hilbert. This is often forgotten by mathematicians in the pursuit of rigour. For example, Dieudonne in his History of Algebraic Topology dismisses the 19th C attempts at this subject as lacking in rigour and calls it 'prehistory' and spends much of his introduction criticising Poincare for his lack of rigour, his imaginative jumps and his assertions without proof and the like, even though it is generally acknowledged he was one of the main founders of the subject. Hindsight is a lovely thing, but its clearly not available to those who preceded us in time.

It's worth setting the Continuum Hypothesis in context. Basically there are two sequences of cardinal numbers we can construct - the aleph and the beth numbers - and the Generalised Continuum Hypothesis states that they match up, for example, the third aleph number equals the third beth number and so on. The Continuum Hypothesis in this language simply says the first aleph number equals the first beth number.

So is it true or is it false? The evidence so far is that we haven't decided yet!

One early bit of evidence for its truth was that it held in Godels constructible universe. But there are other universes where it doesn't hold. This is why some people advocate a plurality of set theories.

This hypothesis should be compared with other major innovations in mathematics: the real numbers, the imaginary numbers, the quaternions and non-Euclidean geometry.

In all of these, there was a fierce debate about their truthfulness and hence their reality. But what really shifted the debate was their usefulness. This is what is lacking in the higher cardinalities. They remain stuck in their little corner of set theory isolated from all the other great mathematical currents.

This will change in the future as mathematicians begin to find them useful in natural ways. And it will be this turn that will help us think about the ontology of cardinals. In fact, this is beginning to happen. Category theory uses constructions like the category of all sets or all groups which strict set theory is dead set against. Thus category theory runs across what are called 'size issues'. Another example, is that Vopenka's Principle which marks out a large large cardinal (as opposed to a small large cardinal!) is naturally implicated in a theorem. For what its worth, the theorem states:

Vopenka's Principle implies all reflective localisations of any infinity category exists.

Localisation is a process of adding inverses. For example, the localisation of integers are the rationals. And the above is talking about the categorical analogue.

Premise 3 of your argument is strictly false. Mathematicians who are not heavily invested in ZFC or even who are but who are as yet unfamiliar with Easton's theorem tend to 'have an intuition' that ℘(A) > ℘(B) whenever A > B. It is intuitive in the quasi-perceptual, not the "I have a hunch," sense, that 2n > 2m when n > m, because we can write out both powersets directly and count their elements, and it is not a hard task to perform transfinite induction over the finite n and establish the relevant principle.

But vs. infinite sets and forcing, it is possible to have 20 = 21, and these can both be equal to 22, 23 ... 2ω ... etc. Moreover, unlike the finite n, ℘(A) for some transfinite A is not only equal to 2A, but 3A, 4A, ... up to AA. By contrast, there are only two cases among the finite n where the basic tetration set over a number is equal to the powerset of that number, viz. for 00 (if you accept the arguments for 00 = 1, btw!) and 22. But 2 by 2 is 4 regardless of whether "by" means addition, multiplication, exponents, tetration, etc. So whatever intuition tells us about finite powersets, it seems ill-equipped to tell us as much about infinite powersets.

Objection: humans do apparently share a range of perceptual types: everyone is at least able to see or hear or taste or ... And if we believe in not Platonic, but Kantian forms, i.e. space and time as formal intuitions, then if our spatiotemporal perception is continuous, then one wonders whether we have something like an intuition of "the" Continuum, so that we can vaguely quasi-perceptually discern that it is not equal to the set of all countable ordinals. It took a long time for people to come up with and accept the notion of different-sized infinities, though, and I confess that had I not had Cantor's zig-zag argument shown to me, I would have been tempted to think that there are more fractions than integers; so my ultra-naive set-theoretic self would have gotten a fairly simple matter quite wrong, on grounds of my deference to my preliminary intuitions.

At best, and only if I knew that the set of ratios of integers was equal in cardinality to the set of integers, but not the set of real numbers, I might've thought, "Hmm, well, we learn about the irrational numbers based on learning about the set of integer ratios, so it's like the set of irrational numbers is the 'conceptual successor' of the other set, so..." in which case my intuition would have favored CH, though.

But so if evolution had instilled ~CH in our minds, then why did it do so? And how further specified an installation are we speaking of? Did it also implant, "20 = ℵ192,315," in anyone instead? Or any other specific such equation? I would suspect that evolution, if it had a "direct hand" in such matters, would still have impinged more on our "sense" of GCH rather than CH. Yet once again, and historically, GCH did not "intuitively occur" immediately to Cantor and others during that initial period, even if CH did, and on top of that Peirce came up with an incorrect argument for the nonexistence of ℶω (though it also came to be known that something similar to Peirce's argument was true, namely that ℶω, or even ℵω, is effectively a 'large cardinal' vs. ZFC minus the replacement scheme).

So you might be asking for an explanation of a nonexistent fact, and it is not clear enough whether you are talking about even a possible fact (aside from, perhaps, bare 'logical' possibility, but almost certainly not, it seems, nomological possibility) that we can at least intelligibly imagine counterfactual explanations of. And I don't know what an analytical (or continental!) philosopher (as opposed to a set theorist or maybe a physicist with a penchant for neurobiology or cognitive science) would be able to do to explain why evolution 'told' us ~CH, even if it had told us this.

From Britannica: "For the 18th-century German philosopher Immanuel Kant, form was a property of mind; he held that form is derived from experience, or, in other words, that it is imposed by the individual on the material object. In his Kritik der reinen Vernunft (1781, 1787; Critique of Pure Reason) Kant identified space and time as the two forms of sensibility, reasoning that, though humans do not experience space and time as such, they cannot experience anything except in space and time. Kant further delimited 12 basic categories that act as structural elements for human understanding."

EDIT: To further illustrate the non-intuitive/counterintuitive qualities of transfinite arithmetic, I will bring up the difference between

0 + ℵ1 + ℵ2 + ...

and

0 × ℵ1 × ℵ2 × ...

König's (set-theoretic) theorem is, vs. its simplest case, that the sum of all the ℵn is smaller than the Cartesian product of all the ℵn. Therefore ℵω0 is greater than ℵω. By contrast, for 𝔠 = the cardinality of the Continuum, 𝔠0 = 𝔠. This entire phenomenon (singular cardinals admitting of a different transfinite arithmetic than regular cardinals) is hardly 'intuitive' in the required sense of 'intuitive' (AFAIK König incorrectly concluded that the Continuum didn't have a cardinality assigned to it among the alephs, to Cantor's renewed consternation at the time).