# Is a fallacy involved in perceiving the solution to a long-standing problem as simple and straightforward?

In university (or, school) we're often presented with a problem, and directly afterwards, the solution is presented. Quite often, the solutions presented for a given problem seem painfully obvious, and I am wondering if I commit a logical fallacy in believing that they are.

The most glaring argument in favor of this is the fact that a lot of intelligent people have spent years on these problems, but when they're presented in class the solution seems simple, as if it would never make sense to suggest another explanation.

I wonder if the "fallacy" would be equivalent of survivorship bias, but for ideas, rather than people? I found this question with an answer that seemed to give a guidance but I couldn't figure out how my "fallacy" fit in the chart.

• Slightly related is the P-NP problem: In short, finding a solution may be not as easy as verifying a solution. Commented Jun 2, 2020 at 10:10

No, there is no "fallacy" in perceiving it this way, the sleight of hand is rather in the presentation. Two types of effects, which often work in concert, conspire to create this perception of "inevitable simplicity" of the solution. I will illustrate with examples from mathematics. In a post Coming up with short “magical” proofs on Math SE a user shared their frustration with one of them:

"As I have spent more and more time studying mathematics, I see more of these "magic" solutions in which some obscure identity or property is pulled out of nowhere and used to facilitate a proof...often, multiple such jumps are used in a proof...and these proofs are meant to be done in an hour or so. I feel discouraged by this, because I don't understand how to do these "magical" things."

But unless one tried their hand on it first, and only read a textbook or class presentation, they may not even notice the work done by such tricks. Textbooks tend to "prepare the ground" beforehand, or at least to smooth out the emergence of a key idea in the course of proof. The real, conceptual, difficulty in coming up with them is swept under the rug and obscured in presentations whose authors are aware of them from the start. But textbooks are often cleaned up imitations of what happens historically. Some initially tough problems are solved by what Grothendieck called the "rising sea", see McLarty, Rising Sea: Grothendieck on simplicity and generality:

"The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it... yet it finally surrounds the resistant substance... [The theorem] is submerged and dissolved by some more or less vast theory, going well beyond the results originally to be established".

The core difficulty in the original problem is in the absence of a framework, a paradigm of concepts and intuitions, that serves as a scaffolding for scaling it. And it recedes as the framework advances. This happened to many Archimedean and Apollonian intricate demonstrations after the development of coordinate geometry, calculus and algebraic methods, or to Gödel's incompleteness proofs within the modern mathematical logic, see Has any 'difficult' proof ever been superseded by a 'simple' one? Peirce talked about theorematic (vs corollarial) proofs as introducing

"something not implied in the conceptions so far gained, which neither the definition of the object of research nor anything yet known about could of themselves suggest, although they give room for it".

But when the paradigm is already established "the conceptions so far gained" make all the difference. Modern expositions tend to present even the original problem as already immersed into a framework designed to "submerge and dissolve" it. Notation and terminology are modernized so that they already by themselves suggest a path forward. "Proper" intuitions are reinforced and contrary ones suppressed or dimissed. What appears as an "obscure identity or property" to the uninitiated often becomes a prominent structural element in this risen sea. The aberration of historical development that results, and the impression of "inevitability" created when such modernizations are taken at face value are well known to historians.

A nice illustration of this is the story of counting infinities, see Is there an alternative to Cantor's cardinalities that makes proper subsets smaller than their sets? Since before Aristotle philosophers and mathematicians alike struggled with comparing and quantifying infinities, and grasping the nature of the continuum. It was not until Cantor at the end of 19th century that the issue was finally settled, more or less. Yet Gödel, writing in What is Cantor’s Continuum Problem? (1947), claimed just the "inevitable simplicity" of Cantor's solution:

"Closer examination, however, shows that Cantor’s definition of infinite numbers really has this character of uniqueness. For whatever “number” as applied to infinite sets might mean, we certainly want it to have the property that the number of objects belonging to some class does not change if, leaving the objects the same, one changes in any way whatsoever their properties or mutual relations (e.g. their colors or their distribution in space)... So there is hardly any choice left but to accept Cantor’s definition of equality between numbers, which can easily be extended to a definition of “greater” and “less” for infinite numbers..."

But it is exactly over definitions of equality, and the paradoxes of infinity they lead to (such as the equation by 1-1 correspondence of infinite wholes to their proper parts), that philosophers and mathematicians argued over for centuries. It is not that the original difficulty has disappeared, but rather that it was shifted to selecting a particular paradigm. A textbook or a class teacher is already committed to a paradigm, the modern one, and inculcates the reader/listener in it step by small step, hence the difficulty becomes invisible. Still, as it turned out pace Gödel, the modern paradigm for infinities and the continuum is not the only possible one.

• This is an absolutely brilliantly thoughtful and articulate answer. Thanx. Commented Jun 5, 2020 at 19:14

The solutions of many problems are actually quite simple when thought about appropriately. Often the means of getting there are quite complex.

Take for example, global warming. The obvious solution as any bright teenager will tell you is to move over to renewable energy. Its clean, sustainable and perpetual for as long as the sun shines - and so as far as the human race is concerned -, forever.

There. Problem solved.

But, of course, the main difficulty lies in the already immense investment in fossil fuel technology over the past four hundred years. And hence the immense lobbying power of those determined to hang onto this power, and I mean political power, that this investment represents.

To get past this is the responsibility of governments as they are not beholden to any private parties or special interests, but to the people themselves, in whose voice they speak, and whose bodies they represent and not merely to those small companies of men and women who form the nucleus of largely unaccountable corporations, no matter how large and powerful they are.

Governments are well aware of the power that they wield, but they don't always have the vision and determination to pursue it. Take for instance the current corona virus epidemic. Given the situation is acute, they have swung into action with various lock down measures.

On the other hand, global warming is a chronic and incremental problem, which is why their response has been incremental. However, this merely makes the problem of tackling this in future a great deal more harder. Tackling it early with preventative measures and a systemic response to the energy needs globally makes a good deal more sense. But this, as I already said, requires vision and determination.

• @Geoffrey Thomas: Can you tell me in what way my answer is 'abusive' or counts as 'spam'? Commented Jun 3, 2020 at 11:26
• @geoffrey Thomas: Or is your problem is that I'm pointing out the truth about the 'largely unaccountable power' of vast corporations? If so - do tell us. And if not, please explain why. As otherwise I am going to take what you did as an abuse of your moderating powers. Is this why you align yourself with corporations? What do they pay you in? Commented Jun 3, 2020 at 11:32