Some have said, "Two heads are better than one." That's true if they are collaborating. If not, the two together may be worse than zero.

Although that's a rhetorical opening paragraph, this is a mathematical question.

What are the algebraic properties of intelligence?

Is Intelligence Additive?

Is intelligence additive and under what conditions?

If we have a software component containing some AI and we duplicate it and aggregate the two, is the new aggregated component twice as smart? Twice as fast? Twice as reliable? Twice as versatile? Twice as accurate?

Under realistic computing conditions, none of these are true in the general or even common case. Yet we imagine that larger systems will be smarter. Why?

Is it Subtractive?

Is intelligence subtractive? If we create intelligence on earth in one location, does that decrease the total intelligence everywhere else? Is there a law of Conservation of Intelligence. Probably not.

Is it Conserved?

Before we consider the idea of conservation of intelligence, consider conservation of information. Do we have more information now, or just massive redundancy because of easy and fast ability to copy data now? When humanity discovers something, does it forget something else? How would we know if that were true, since we forgot?

This paradox is important to artificial intelligence.

Some Things We Know

It is important to know exactly what kind of hill we think we are climbing. Here's what we do know.

We know that accuracy and reliability can be in conflict. A guess is sometimes more reliable and less accurate, where as an answer with six significant digits can't always be trusted, as in the case of applying Newtonian physics to the orbit of Mercury.

We intuitively know, when we encounter success, to keep it and perhaps replicate it. We manufacture designs that have been proven to work.

We think intelligence is useful, and we can show examples that are convincing. I'm convinced. We want to manufacture that too, yet we don't know if twice the intelligence is twice as good or even what twice the intelligence means in the concrete realm of a computer program.

Comparative Intelligence

Although we don't know what, "More intelligent," means in every condition of comparison or agree on who or what is more intelligent in every circumstance, there is more that we do know. Some of what we know can help us move toward understanding the algebraic properties of intelligence.

A smart entity in one place is not smart in another. This is true of learning. Many write and speak of general intelligence, yet universal intelligence isn't even realized in humans. One would not ask Linus Torvalds to write a song for a television commercial, and one wouldn't ask Christopher Nolan to find a better way to automatically taper the stochastic component in a stochastic gradient descent strategy.

We know that a person placed in one job will get it immediately and if placed in another many not in their life time ever do it well.

That's one key. Intelligence appears to be linked to the type of environmental challenge. As much as general intelligence is discussed as something we hope AI research to achieve, it may never be realized.

Some have said, "You cannot prove the existence of God," which is just as easy to say about general intelligence. There are sometimes no proofs for infinite cases and they cannot be measured for the purpose of proof of concept. Omniscience and omnipotence may forever be outside the scope of human endeavors.

Parallelism is Proven

We do know that if we have a large set of data to process and we detect a bottleneck in the process, we can scale the components of the bottleneck, and in some cases, if done with an understanding of the process and the cause of the bottleneck, produce close to double the throughput by intelligently doubling the computing resources.

In that case, data throughput can scale. However, that may or many not be the case with intelligence. That brings us back to the question of whether intelligence is additive.

Potentially Important Hint

We know this too. If we have adversarial intelligence, the competition may, under certain circumstances, lead to improvements. The intelligence of adversaries may tune one another. Although the competition is adversarial, there is a symbiotic element, and two competitors can be friends outside of the competition. We see this in school as students compete for the high GPAs. This has been demonstrated in the success of simple generative adversarial networks too.

We also know that adversarial relationships are inherently limited. We know that without collaboration, the formation of partners and teams, some things don't occur at all or become mutually destructive. John Nash broadly defined a mathematics of economic equilibria that form in what Morgenstern and von Neumann defined as non-zero sum games. We see in history the effectiveness, the legitimacy, and sometimes the elegance of collaboration.

What would George Westinghouse have accomplished without Tesla? Would China and the U.S. be largely on the same page today (in spite of their grossly different view of political structure) had Nixon and Kissinger not collaborated and then initiated more collaboration with Mao Tse-tung and Chou En-lai, who had also been collaborating?

Competitive interaction within the biosphere helped e. coli evolve. It helped jackals evolve too, but symbiosis had much to do with the progress of both species. Jackals couldn't digest their food efficiently without the e. coli bacteria that benefits from the hunting skills of the jackals. Both organisms may have benefited from survival of the fittest, yet without symbiotic relationships, both life forms would have been considerably diminished.

What Conditions Determine the Algebraic Properties?

When does what algebraic operation on intelligence model reality?

Perhaps intelligence is not a property at all. Perhaps intelligence is an umbrella term for a set of much more precise quantities that behave reliably. In scientific history, it would not be the first time. Aristotle spoke of attraction, but physicists now know about gravity, electrostatic and electromagnetic forces, bonding, and other effects that can be modeled with precision.

The work of Google and CalTech on PAC Learning is one of the systematic approaches to the phenomena of learning. Such work steps toward the ability to work with AI metrics algebraically.

Quantifying Intelligence in $\mathbb{R}^N$

When we talk about more intelligence, we infer that intelligence is a quantity. Here and elsewhere, I've drawn attention to the dysfunctional side of representing intelligence as a scalar in $\mathbb{R}^1$. I've proposed that intelligence must, at the very least, be represented as vector in $\mathbb{R}^N$, where N is at least twenty.

Whether intelligence is a vector with dozens of features (dimensions) or not, how can we work with it as a quantity? Or are we wrong to think of it as quantitative? If it is a quantity, must it always be relative to a specific set of problems no matter how large we grow a set of intelligent capabilities through continued AI development?

What are the algebraic properties of intelligence?

Response to Comments

Although Perlovsky's work is interesting, the Wikipedia blogging in general and in this case is not. Perlovsky does delve into the nature of intelligence, but does not consider its algebraic properties (as in abstract algebra) such as Morgenstern and von Neumann did with game theory, which is the central theme in this question.


There are no intrinsic algebraic properties of intelligence because it can be manifest in different ways.

For an example where you can measure the algebraic properties, I like the definition of the current AI Spring as "cheap intelligence". Deep neural networks are demonstrably more intelligent that previous systems when measured against things like

  1. recognising words from sound
  2. recognising objects in images.

The way they employ algebra varies but fundamentally have linear transformations (such as convolutions) and nonlinear transformations (such as ReLU).

Read more Convolutions in MS Excel! See this excellent tutorial on how to calculate convolutions by Jeremy Howard of fast.ai. He also shows the ReLU function


Then I'm saying "Neural Networks" I mean Artificial Neural Networks.

Is Intelligence Additive?

Is it Subtractive?

No to both. Neural Networks show that quite clearly.

Is it Conserved?

In some sense. Probably. Intelligence is associated with algorithmic entropy and we know that normal entropy conserved for reversible process. Again we know that Neural Network computation could be made reversible. The gist of it:

Some important properties of intelligence conserved under under some broad class of transformations.

Quantifying Intelligence in R^n

R^n likely too narrow space for it. Intelligence as we see it more likely reside in the space of transformations, which by its nature is functional space. Instead of quantifying intelligence it could be more practical to find invariants of intelligent processes, analogous to topological invariants.


Intelligence according to Wikipedia is

Intelligence has been defined in many ways to include the capacity for logic, understanding, self-awareness, learning, emotional knowledge, reasoning, planning, creativity, and problem solving.

Intelligence is the state where the above qualities work in harmony to solve a problem and have a level of abstraction. Intelligence could be not measured as of now. But it could be analysed by giving a problem to it and measuring its accuracy and error margin.

  • $\begingroup$ I tried to say that Intelligence is a state and not a quantity which could be measured. $\endgroup$ – Shubham Panchal Oct 13 '18 at 15:11
  • $\begingroup$ Can one tell how happy or sad you are now in mathematical units ? The same problem is with Intelligence. It exists but cannot be quantified. $\endgroup$ – Shubham Panchal Oct 13 '18 at 15:30
  • $\begingroup$ No it is not supernatural. It is just the unexplored part of science ! $\endgroup$ – Shubham Panchal Oct 13 '18 at 15:37

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