Higher education, at least in some areas, is often extremely competitive, with vast disparities between the quality and quantity of knowledge transferred to people within different institutions and strata of educational attainment. This post presents a model of higher (and possibly all) education, and makes the case that it implies that it would be massively beneficial to increase the total amount of education available in certain areas and reduce this inequality. I also try to show that the analogy renders some of the most widely mentioned arguments that universities should be highly selective inapplicable to the real thing.

Analogy: The higher education system as a rocket engine

In this analogy, a civilization, society or country (or the human species) is likened to a rocket; its goal is to carry a certain amount of something, its payload, into orbit, or to another planet. It is driven by its engines, which represent various institutions, and in this post will largely be taken to refer to educational ones, in particular universities. 

Because of the immense amount of energy required for a rocket to reach orbit, or go beyond, it must carry a lot of propellant, and this propellant must carry a lot of chemical potential energy. Unfortunately, the weight of this propellant scales with how much of it there is, so it is imperative to maximize the proportion of the rocket's mass, including that of the satellite, or whatever else it carries, that this propellant constitutes, so as to maximize the entire assembly's energy density.  Adding more propellant only helps improve performance if it doesn't require the addition of proportionally more other mass. 

Another way to maximize performance is naturally to maximize the energy density of the propellant itself, leading to a concept known as specific impulse. This can be defined as the ratio of the amount of momentum the rocket gains by burning a certain amount of its propellant, to the mass of this propellant. This is equivalent to the average backwards component of the velocity of the propellant leaving the engine, relative to the rocket. Since different engines use different kinds of propellant, specific impulse is often referred to as though it is a property of the engine itself, its efficiency; clearly, if a certain amount of hydrogen and oxygen combine to explode with more energy and speed than the same amount of oxygen combined with another kind of fuel, an engine which consumes hydrogen has to consume less propellant than another engine would to impart the same amount of momentum to the rocket, making it seem as though it's more efficient. Importantly, it is not, at least not necessarily.

The hydrogen engine is an analogue of an elite university which only selects the most strenuously refined students/hydrogen atoms; the impressive rate at which it accelerates those it accepts to great achievement/speeds, and amount of intellectual contribution/momentum it extracts from each one lead people to conclude that it must be incredibly efficient, when in reality it is not. It simply uses a more energy/ability rich fuel/student body.  Of course, it could be argued that the lack of actual efficiency matters less than the specific impulse, as it is the specific impulse which dictates the amount of fuel necessary to gather a given amount of momentum, but this is one area where the analogy breaks down, because the total volume of innovation necessary to change a civilization in a particular way does not scale as anything resembling a linear function of the number of its participants.  It would also be immoral to exclude them from civilization. 

^[1] 

Another way to improve the efficiency of a rocket is to actually make its engines more efficient. This corresponds to improving pedagogy and other aspects of educational institutions so as to impart more, more important and useful knowledge to a given number of people more rapidly and with the use of fewer resources. As in the case of a rocket engine, optimizing efficiency is ideal by definition, all else being equal. However, because of the fact that certain fuels, for example refined kerosene, contain less energy than others, there is a theoretical upper bound on the specific impulse that any engine which uses them can possibly reach, even by being 100% efficient, which is to say using all of the energy within its propellant for propulsion. While our understanding of human intelligence and learning ability is not sufficient to provide a similar upper bound on what humans can achieve, it seems likely that, at least within an environment in which humans remain the most intelligent beings, which approximately resembles the current state of the world, they are unable to exceed a certain level of intellectual growth within their lifetime even subject to almost arbitrarily efficient teaching methods not involving significantly altering who they are. In other words, the array of possibly comprehensible things for any particular human, within constraints in which they could reasonably be considered to be the same person, seems to be at least significantly determined by genetic and other factors which take their effect at a very young age. This places a 'soft' upper bound on the possible performance of an education system which is dependent on the people it educates.  

In fact, it is often neither the specific impulse or actual energy efficiency of rocket engines which has the greatest impact on the overall efficiency of the entire rocket. This is because of a phenomenon known as gravity drag, which is the detrimental effect of the requirement to overcome the acceleration due to gravity merely to hover without accelerating,^[2] which creates additional urgency to escape the planet's gravitational field. One way of understanding this is to observe what happens to the momentum opposing that of the recently expelled propellant. In deep space, or in orbit, this momentum adds to that of the rocket, including the rest of the propellant still on board, increasing the energy it carries by combining kinetic with the already present chemical potential energy, which explains why rockets with constant thrust become increasingly powerful and efficient as they accelerate away from a stationary observer. However, when the rocket is trapped within the gravitational well of a celestial body like a planet or a moon, instead of, or as well as, adding to its kinetic energy, the impulse generated by the engines is transferred to the planet or moon by gravitational attraction.  ^[3]

The only obvious way to minimize the detrimental consequences of this phenomenon is to escape from the gravitational well quickly. At this point specific impulse once again comes to dominate the factors contributing to the rocket's performance, which is why upper-stage engines tend to use hydrogen. ^[4]

This requires the thrust:weight ration of the entire rocket to be increased, which in turn requires the early-stage engines to be powerful.  In the context of the education system, as in that of a rocket engine, the most reliable and scalable way to achieve this is to increase the total volume of propellant/students consumed in a given amount of time. Even at the expense of the energy efficiency and thrust:weight ratio of individual engines, doing this is still likely to increase the efficiency of the entire rocket.

I claim that this is (among) the simplest way(s), within the constraints associated with remaining recognizable to current humans, to improve the performance of the education system and therefore benefit civilization, as its effects are likely to compound on one another as they propagate through the world, at least assuming that humans remain relevant.  However, as with a rocket engine, it is never unimportant to pay attention to efficiency and thrust: weight ratio; if these things approach 0, they can offset arbitrary gains due to 'flow rate' simply by doing so.  While there are many humans capable of learning with minimal infrastructure surrounding them, and a much smaller number capable of doing so efficiently, because of the importance of this efficiency, it is probably, excepting AI, necessary to attempt to extend existing universities to accommodate larger numbers of people as opposed to removing infrastructure or increasing student : teacher ratios alone.  In particular, I think the efficiency gains due to being able to ask questions to an individual human who answers them to the point of exhausting curiosity, are too great to forgo, at least in some areas.   

One way to increase the capacity of civilization to provide this kind of education^[5]would be to redistribute resources to reduce the disparity between different universities. 

Although there is no type of spacecraft, as far as I am aware, analogous to the modern education system in the variety of engines/universities, we can imagine that there was.

This would be a rocket with many engines on a single stage, perhaps the only one required for it to reach orbit. Some of them might consume hydrogen, while others might consume hydrocarbon fuels. If we assume that the fuel for each of these engines must be combined with oxygen to explode, the question of how to distribute this oxygen between them arises. Ideally, each engine would receive exactly as much oxygen as would be necessary to react with all of its fuel, and no more, preventing any of the oxygen, or fuel, from being wasted while extracting the maximum amount of energy and momentum. However, since our bizarre spacecraft is analogous to the modern higher education system, this is not actually the case; all of the hydrogen is burnt, but the hydrogen engines receive far too much oxygen so that each hydrogen atom would need to react with many oxygen atoms to exhaust them, which it cannot do, while there is insufficient oxygen for the kerosene engines to achieve their maximum specific impulse, however efficient they are, so some kerosene is wasted. (In the analogy, oxygen is equivalent to resources and staff at universities.) 

Clearly this situation is suboptimal, and the solution is to allocate the oxygen/resources more evenly, although perhaps not at a 1:1 ratio. If you doubt this, I would suggest doing research into the way that resources are apportioned to universities at different levels of prestige and competition.  The benefits of doing this could be colossal and qualitative, in the same kind of way that the benefits of improving a rocket can be qualitative; perhaps it couldn't actually deliver any payload into orbit before, and it now can. 

In the higher education system, I expect it would take the form of increasing the swathe of universities which taught a complete curriculum, as well as evening out the distribution of staff. 

Responses to potential objections:

Optimality of the current system(s)

The distribution of teaching resources really is optimal; just as in gradient descent, the parameters which correspond to dimensions of steepest inclination are exactly those which it is optimal to decrease(or increase) most rapidly, the most capable and intelligent students can make optimal use of the educational resources required to teach them, and convert them into the most significant benefits for their society/civilization/species.  

My response to this is that, in the absence of brain augmentation, there is obviously a kind of  "saturation point", beyond which no human could learn faster even if an arbitrary number of resources were expended on their education. In addition, extremely intelligent humans tend to be able to learn much more efficiently alone.

Alternatively, someone might be universities to argue that institutions are limited by the number of available students who would be capable of understanding what they teach. This explains the radical differences between the curricula at different universities.  I think this is obviously false. It is certainly possible for most applicants to an elite university, for example among the top 16 universities in the world as measured by the depth and breadth of their curriculum, to learn General Relativity in that level of depth as an undergraduate student, but the vast majority of them do not because they fail to obtain a place and go to a worse university where it is not taught, or taught to a lesser depth. One type of evidence for this is that the tests used to select for this capacity to understand the material would need to be phenomenally accurate to account for the degree of homogeneity among professors and people who win prestigious awards with respect to the universities where they gained their undergraduate degree. See this link for an example : https://en.everybodywiki.com/List_of_Fields_Medal_winners_by_university_affiliation   Note that a single university in France has produced 13 alumni who have gone on to win the fields medal, more than 1/5 of the total number of winners in the entire world , 60.  This suggests that if there were more equivalent universities in other parts of the world, possibly along with similar mathematical education in high/secondary schools, there would be many more fields medals awarded ( this would require the number awarded to be increased in proportion to the amount of additional fields - medal - level mathematics being done) .

Signalling model

The education system acts as a kind of 'human centrifuge' which stratifies people in a legible way which allows other institutions to use this to select them, and because it exists on such a large scale, it does this much more efficiently than other institutions could on their own. In addition, this is a good thing. Reducing the disparity between universities would reduce students' incentives to concentrate themselves at the few most selective ones, which would make this mechanism of selection less effective. 

My response to this would be that, as others have suggested, assessment and accreditation could be separated from teaching institutions. There are multiple reasons why this would be a good thing, one of which is that it would move the steep gradients of competition away from universities and place pressure on other institutions such as corporations to ascertain which assessors most accurately measured the traits they found to be most valuable in their employees, which would in turn create an incentive for these assessment boards to become better at detecting important qualities and knowledge, as opposed to ensuring that they reflected well on the educational institutions of which they are currently a part or connected.

AI 

As AI becomes continually more intelligent, the importance of educating humans declines proportionally, so that this entire debate is irrelevant. Furthermore, AI will soon surpass human teachers in various ways, as it already has in the domain of knowledge apart from understanding, and this will make any required improvement in education possible, to the extent it's still necessary. This renders structural change as described in this post unhelpful. 

My response is that the post was written under the assumption that AI does not vastly outstrip humans in general intelligence, although this seems likely to happen, because even if it does, it makes sense to improve, or at least understand the current deficiencies within, the education system while humans remain relevant and can contribute to important things. I expect that explanation of a concept to a sufficient degree of depth to convey deep understanding to another human requires that the entity doing the explaining understands the concept in question deeply themselves, and that understanding a wide range of concepts to this level of depth is a central component of intelligence.  This means that until AI reaches superhuman intelligence, at which point humans will become obsolete, they will retain certain advantages important for exposition over AI. 

1. ^{^}

   Having written the rest of the post as I write this foot/sidenote, I can say that the analogy can be modified, or more charitably extended to accommodate these things. 

2. ^{^}

   This paragraph explains the phenomenon within Newtonian physics. In general relativity, it would be more correct to say (as it presumably is in reality as well) that the acceleration due to gravity the requirement to overcome gravity is necessary to hover without accelerating distancing oneself from the planet's surface; the hovering maneuver requires acceleration because worldlines parallel to the planet's surface in spacetime are not straight lines, but rather equidistant curves which continuously bend away from the centre of the planet.

3. ^{^}

   I have moved the paragraph I wrote attempting to explain this within General Relativity to this note because I am not sure whether it would be appropriate to include two different explanations of the same thing from within two separate theories, if readers are only interested in how it relates to education etc.   Please comment if you think it should be moved back to the main body of the post.

   In General Relativity , local conservation of momentum bends the rocket's trajectory away from the planet's centre, but this gain in velocity is needed simply to counteract the rocket's tendency to converge with the planet.  One way to explain why this is would be to point out that the curvature breaks the equivalence between the rocket's (local) velocity, and the (global) rate at which its distance from the planet increases, which would require a global notion of a parallelism. This means that it's entirely possible for the rocket to deviate away from the planet without distancing itself from it at all! 

    

4. ^{^}

   Upon reflection, I'm not sure whether gravity drag and the consequent necessity of escaping the gravitational well have direct analogues in human civilization... perhaps certain, large scale endeavours, such as interpretability AI research , involve counteracting concurrent progress in the opposite direction, such as AI becoming more complex, and could be used as an example here. This seems like a strong argument in favour of my position.  On the other hand, I'm not even sure that these are soluble problems, making actually 'achieving orbit' a futile goal to pursue. Given this, I haven't included such examples in my analogy, but note that the value of total thrust:weight ratio as opposed to that of the engines does have a direct analogue. I also strongly suspect that there are other analogues and ways for returns on investment into education to be nonlinear (in a good way) , I just don't understand them well enough to say.

5. ^{^}

   In the context of, for example, physics, I think this consists of satisfying a few key criteria: 

   1. Depth of the knowledge transferred to students, which would involve, for example, a complete explanation of both the conceptual components of General Relativity ( the light cone and everything it entails from special relativity, the general equivalence principle and the way this relates to Newtonian gravitation ), and the mathematical description of these things ( Minkowskian geometry, curvature and the relationship between tensors and vectors)
   2. Breadth of knowledge transferred to students, which would require them to be introduced to all of Quantum Mechanics, and Classical Mechanics, and Electromagnetism, and Thermodynamics from both theoretical and experimental perspectives
   3. Mathematical rigour which would be sufficient for the students to understand the mathematics of , for example, quantum mechanics without any knowledge that it describes physical reality, with the exception of the rare areas where this knowledge is not available ( I have been informed by the internet that this is true of path integrals over possible trajectories taken by particles in Quantum Field Theory, but I don't understand this area myself and have never encountered an area of physics whose underlying mathematical framework was not completely understood while studying in an official capacity. Nevertheless, much of it is absent from university physics courses I took.) I should point out here that this doesn't require everything to be introduced in terms of axioms, as would be the case in a mathematics course, for the simple reason that some axioms are intuitively obvious; for example, Euclidean geometry is sufficiently intuitive that it would not need to be introduced this way, but Riemannian geometry is not, and would therefore require significantly more explanation of its mathematical formulation.

   4. Conceptual rigour of a level which would allow a student to losslessly compress their knowledge into their head/brain in such a way that they themselves could explain it to another student so as to provide the same level of understanding, assuming they were a skillful expositor. In other words, not leaving g a p s in their understanding.