# Is the singularity concept mathematically flawed?

In Comes IQ

When the concept of Intelligence Quotient arose it was based on this approximation.

Each human being has a number that quantifies their intelligence relative to a fixed norm, and, although this number may vary from testing to testing and over the course of years, it varies only within a small statistical deviation and is essentially determined at meiosis, when the DNA is assembled for that human.

Experiments and studies supported the validity of IQ as a measure of intelligence. Others refuted the concept. It was ethically questioned on the ground that it fostered intellectual elitism. There were legislative discussions regarding discrimination on the basis of IQ.

Andrew Niccol's brilliant 1997 motion picture, Gattica, starring Ethan Hawke, Uma Thurman, and Jude Law, dramatically examined the question of genetic determinism.

Research overturned the elitist belief that IQ is purely a mater of pedigree. The current evidence-based perspective is that from six months old to adolescence, nurture dominates over genetics.

Nonetheless, during the 20th Century, IQ rose to become a common household term. People knew, for example, that Albert Einstein was supposed to have had an IQ greater than 200. How that was calculated, since he never took an IQ test, few have stopped to scrutinize.

Unlike the single number used to gauge intelligence, college boards gauge achievement using a vector of numbers.

• Command of evidence
• Vocabulary
• Ability to express ideas
• Following of language conventions
• Mathematics
• Problem solving
• Essay writing

It is interesting that a set of numbers are used to gauge academic achievement yet all the complex genetic and educational factors that impact a person's intelligence are rolled up into a one-dimensional number.

• Within the complex genetics, circuitry, and chemistry of the human brain is there some single factor that determines the intelligence of an individual?
• Does one factor eclipse and render irrelevant all other factors that are known to affect how smart a person is?
• Does each person have a glass ceiling on personal growth and education that is locked in by their mid-teens?
• Is this what people really believe?

Today, Peter Norvig's word frequency table places IQ among words like mailbox and defect at the 96.9 percentile position in terms of frequency of use. This linguistic fact is strong evidence that our culture responds to these questions with, "We buy IQ."

In Comes The Singularity

Now the concept of The Singularity can be reduced to this.

Humans will work on artificial intelligence diligently, causing the peak level of intelligence of an artificial variety to increase. Once the intelligence of an artificial variety exceeds the intelligence of humanity, the artificial intelligence will gain control over humans to guarantee the perpetuation of its own existence and continued self-development.

The Singularity concept relies on a very specific pair of propositions. (Using pseudonyms)

The intelligences of Abby and Beth are fully represented as scalars (1-D) not vectors (multidimensional) and therefore can be directly compared.

If the intelligence of Abby is greater than the intelligence of Beth, Abby will gain control over Beth and exploit her.

In Comes Proof of the Limits of Proof

The dominant enlightenment philosophies of humanism and determinism, that the intellectual pursuits of men will grow to dominate all things, were challenged in the early 20th Century by the work of two people.

• Heisenberg proved, and there has been no credible refutation, that one cannot accurately measure two determining extensible quantities of a particle at the same time and therefore any mathematical treatment of Bohr's model of the atom must be based on probabilities, not the certainty of mechanical phenomena.
• Gödel proved two things (a) That a formal mathematical system cannot be relied upon to indicate the truth or falsehood of every statement that can be made formally, and, more importantly, (b) no formal system can, though its own formalisms, establish its own consistency via proof.1

This uncertainty appears in linguistic expressions like, "Ricky doesn't test well," when Ricky's test score is below expectation (again using a pseudonym). There is scientific validity to these sayings because neural activity cannot be measured in natura (in natural life). We must bring the subject into the lab to measure their neural activity, which disturbs the neural activity of the subject under study, just as the electron in Heisenburg's thought experiment.

Applying Gödel's thinking, human intelligence cannot construct a formal system to determine inconclusively that human intelligence is can be expressed as a scalar, can be expressed as a vector, or even exists at all.

We cannot, via intelligence, formally prove that intelligence is self-consistent.

In Comes Genetics

Research has recently revealed that, of 12,000,000 human single-nucleotide polymorphisms (SNPs) examined, 336 have been found that significantly correlate with human intelligence, and those 336 SNPs implicate 22 independent genes.2, 3 It is also likely that additional intelligence related genes will be discovered as research continues.

Even if there are no further independent genetic determinants of human intelligence found, twenty-two genetically distinct, independent degrees of freedom define the variable features of human intelligence. An important conclusion then falls directly from this research.

A scalar, such as IQ, cannot possibly describe the permutations of genes that human intelligence.

At least a 22-dimension vector is required to represent the multifaceted features of an individual's intelligence.

Putting the Proof Together

Although the MAGNITUDE of a vector can be greater than the MAGNITUDE of another vector (without first taking the absolute value of the vector) it is mathematical nonsense to say vector X (of 22 dimensions) is greater than vector Y (of 22 dimensions).

We also lack any evidence-based reason to apply an absolute value to the vector. A pure root mean squared calculation (RMS) would mean that human intelligence is a Cartesian hyper-cube of 22 equally weighted traits. There is no evidence to validate the idea that the mathematical inventions of Descartes should be applied to genetic traits.4

All these traits are situational anyway, arising out of specific environmental challenges during evolution. For instance, trait number 9 may produce an intelligent response in one scenario and trait numbers 11 and 17 together may produce an intelligent response in another scenario.

So without creating some basis for mapping the traits to the scenarios where they are important in survival, problem solving, or human advancement, it is NOT MATHEMATICALLY SENSIBLE to say that Abby is or is not smarter than Beth (using the previous pseudonyms).

Neither can we say that a machine named Polycharp is smarter than Abby. Consequentially, what is described above as The Singularity cannot possibly be singular.

We have only three logical possibilities. Only one can be true, but which one?

• The genetic research is flawed.
• The concept of The Singularity is flawed.
• The logical inference laid out in this question is flawed.

The Unpredictability of Cybernetics

The proposal that has been suggested in the literature is that the detailed workings of human intelligence may not fit into the network size of the human cerebral cortex. That is a reasonable line of scrutiny. Just as particle physics and the complexities of the biosphere may always require computer analysis and be visible only through the lens of statistics, so may be the case with human cognition.

Adding the work of Heisenberg and Gödel and it only gets worse. Even a computer model of the twenty-two or more dimensions of human intelligence may be as elusive as physical quanta, further confining the evaluation of intelligence to statistical treatment.

These factors place the interplay between human beings and machines (cybernetics) firmly within the realm of the unpredictable.

Super-intelligence May Face the Same

It may be that some higher intelligence that exists in the universe or some superior artificial intelligence we invent can both fathom human intelligence and predict the course of cybernetics. However, that greater intelligence may eventually be confronted with the same issue. Such beings may not have the intellectual capacity to fully fathom their own intelligence or predict their own cybernetic destiny.

Returning Back to the Main Question

Regardless of the predictability of cybernetics, it is reasonable to ask,

"Is the singularity concept mathematically flawed?"

Footnotes

[4] Assuming the twenty-two or more traits involved in human intelligence should be aggregated via an RMS function would be as subjective (and ridiculous) as applying RMS thusly: F2 = b2 + b2, where F is fun, b is blonde, and B is blue-eyed, because fashion magazines sold contact lenses and hair color products to a primarily brown-haired, brown-eyed population by depicting blue-eyed blonds having fun.

• I do share some of the opinions about "The Singularity" with this post, but as phrased those parts of the question are inviting just that, opinions. – Neil Slater Jul 30 '18 at 7:18
• I have assumed this SE different from other SEs and that it allows broad questions, now that I have taken the tour I understand that questions that result in discussion rather than answers are discouraged ( or should I say not allowed). Am I correct ? – papabiceps Jul 30 '18 at 7:23
• @papabiceps not only concepts of AI are highly opinionated, answering so many questions by a single person is impossible in any kind of site – DuttaA Jul 30 '18 at 7:30
• @NeilSlater: yes, better to Meta (but in Meta there are already a lot of questions about this site scope and the contradiction between its definition and real content). It is difficult write fact based answers to "conceptual questions about life and challenges in a world where cognitive functions can be mimicked in a purely digital environment", specially when this world is not yet reached or it is only in specific areas with strong limits. – pasaba por aqui Jul 30 '18 at 8:34
• You have completely ignored the communal factor in Intelligence..Genetics only provides the brain machine, dissemination of environment Probably depends on a lot more factors other than brain – DuttaA Jul 30 '18 at 10:58

The logical inference in this question is flawed[?]

I think that is the case here, although I think it is mainly the starting axioms, plus the assumption that there is a sound logical argument for The Singularity that can be attacked. I would take the position that there is no such argument, that the concept of The Singularity is a rough outline and thought experiment only, and as such it is not really possible to refute it so directly.

Some of the inference seems solid, and I think your insight into vector versus scalar qualities of a complex system is an important factor in AI research in general. When researchers talk about "narrow AI", then one way to frame that is "AI with a singular measurable quality that we can easily show is optimised" - although even then there are ways to boil down complex systems into singular values, as used by RL and genetic algorithm approaches. Ultimately, if a system generates new "improved" versions of itself then an A-life approach can be taken where the measure is purely survival (although of course that does not guarantee any specific measure of intelligence will be high)

However, the relationships between your observations and how proponents of The Singularity view the logic behind it are mismatched. I cover genetics below, but for the other headline items:

In Comes The Singularity

Your characterisation of The Singularity does not match any other version of it that I have seen. There is no need to involve specific measures of "general intelligence", such as IQ. The strong focus on IQ weakens your argument, as although there is evidence both for and against a general intelligence factor (which IQ purports to measure), the proposed mechanisms for The Singularity are not so heavily focused on IQ, but on solving the problems in creating an AGI.

To what degree the "General" in Artificial General Intelligence may be represented by a single number is not a key argument for The Singularity happening - instead it is kind of assumed this will be solved during development. That is a potential weakness in arguments that support The Singularity happening, but not because anyone claims anything special about IQ.

In Comes Proof of Limits

These are important theorems that all systems, including any real or imaginary AI, are limited by. No AI, however advanced, will be able to solve the Halting Problem efficiently, it will remain an NP-hard problem. However, a super-fast AI might be able understand and solve difficult halting-like problems faster than any other, purely by brute-forcing on a system which has vast computational power.

In addition, it is very likely that an AI will not be able to:

• Transport items faster than the speed of light

• Break the second law of thermodynamics

• Time travel into the past

Some of the proponents of The Singularity might suggest that if, despite our best theories of physics, these things are in fact possible, that a suitably advanced intelligence could through theorising and experimentation, figure out how to do these things. However, extrapolating "super intelligence" -> "can magically break laws of physics" cannot be taken seriously as a theory. It is science fiction or wishful thinking. Maybe it will come to pass, but these things are currently in the same place as alchemy. Interestingly, we can now perform one key goal of alchemy - transmuting base metals to gold - but understand that it is not worthwhile. When extrapolating into the unknown, this kind of outcome for AGI is also entirely possible.

Current physical theories do set an ultimate limit on knowledge, measuring collecting and processing data, based on thermodynamics of computing systems. Physical limits on computation are highly relevant to computer systems acting intelligently. However, those limits are very high compared to computing power assumed to be required for human brains, so are not practical limits that would prevent AIs with capabilities that outstrip human-kind in all endeavours.

The genetic research is flawed[?]

I have no reason to suspect that the genetic research is flawed. However, it is mostly irrelevant to arguments about AI. Genes are expressed in proteins, and those proteins have specific tasks. A nerve protein might have better binding capabilities to neurotransmitters, or affect development and folding of brain tissue etc. This doesn't link directly to your idea of a vector describing intelligence, as the raw changes due to genes won't map to how we measure intelligence, even when taking a multiple intelligences model of capability.

One thing I would like to address is the focus on genetic determinism. Although there are many candidate genes that (weakly) correlate with measured IQ, there are also many studies that show various environmental factors have an impact. These vary from chemical and physical factors (exposure to lead, birth by caesarian section), to interventions in mental stimuli and education. Early socialisation is quite critical to normal mental development including measurable IQ in humans. I don't think this is hugely relevant either way to arguments about AI though, since DNA roughly corresponds to AI system design by analogy; both biological and artificial systems also require environmental input to learn from.

The concept of The Singularity is flawed[?]

I would say it has many possible flaws. However, I think your arguments hit wide of the mark, as you have taken a "straw man" version of some of the arguments in support of The Singularity. As a result, the arguments presented in the question are unlikely to sway someone who has followed a different set of arguments for The Singularity.

The basic arguments for The Singularity are not formal, or mathematical in nature. As such, it is not really possible to find a strong logical flaw in them. However, there is a lot of conjecture and extrapolation involved.

The argument chain you need to address works like this:

• There is no known reason why an artificial general intelligence (AGI) cannot be designed and built. Such an AGI would be capable of theorising and learning through experience and experimentation.

• There is no known reason why an AGI could not design and build other AGIs

• At some "tipping point" in capability, an AGI should be capable of building another AGI with greater capability than itself. Note that this is capability of building other AGIs, a specific goal, and more amenable to a single-value scoring system than more contentious measurements of "intelligence". There is a subtext that this trait is linked to general measures of intelligence (otherwise the runaway process is more similar to the Grey Goo or Paperclip Maximizer) which is rational as the AGI needs to be able to research and theorise about improvements to itself in order to express this capability.

• This "can build better than itself" trait continues to exhibit in future generations, leading to exponential growth in capability.

• At some point on the growth curve, the AGI's capabilities will exceed humankind's.

• It is not really possible to predict what will happen beyond that point, because it will be the purview of an intelligence that we cannot comprehend. This point is The Singularity, because like its namesake in physics, even this loose theory breaks down and is not able to make predictions.

This last bullet-point is where a lot of magical thinking can appear, as people strive to imagine what a better-than-human intelligence might achieve, but there is no reason to expect anything in particular. The process could stop at any limiting point, and the only thing we know for certain is that it is possible for a general intelligent system to operate at "human genius" level, because we have examples of human geniuses - including the original architects of computing (Turing, Von Neumann et al) so it is clearly possible for an entity to be that intelligent in a way that relates to building other computing systems. That includes biological systems of course, whether we can build or bootstrap to that level is not clear, but those intelligences clearly existed.

This outline of the process is supported by a number of observations of technical advancements already made. Key amongst them is Moore's Law, which shows an exponential growth in raw computing capacity stretching over several decades. Some of this growth is fuelled by using current machines to aid in the design of better machines (it is hard to say whether this is necessary, but it is what has happened). That can be viewed as an embryonic version of The Singularity process already happening, or more weakly, as a working and relevant analogy to how it could work.

I think that arguments to show flaws in The Singularity can attack assumptions in any outlined steps (be aware that this is my interpretation, and thus also vulnerable to being a "straw man" when used in a discussion). The trouble of presenting a solid argument either way, is that most of it is conjecture around capabilities of an AGI - whilst we are not able to properly define what that really is, it leaves the door open to arguments and counter arguments that cannot really address the issue.

• @DouglasDaseeco: Some science fiction turned out to be quite predictive. Arthur C Clarke springs to mind. I don't think you will find anyone to debate with here, and I don't think that such a debate would be meaningful or productive. Assuming your characterisation of The Singularity as religion is correct then witness any debate between evolutionist and creationist . . . adherents to either side of such a debate don't actually change their mind, they just entrench. – Neil Slater Aug 12 '18 at 12:11
• To continue, I don't "buy in" to The Singularity. Any cult-like elements of it also seem distasteful. But I cannot declare the whole thing, lock, stock and barrel, as a complete fairy tale. Predicting the future is hard. Perhaps humans will end up creating super-smart machines that then go on to build things beyond our comprehension . . . no-one yet knows the scale that intelligent agents work on, and where we are on that scale. – Neil Slater Aug 12 '18 at 12:15

From the perspective of game theory, intelligence is the ability to win a game. How intelligence was produced isn't important. It can be done with DNA, a biological computer, an algorithm, human brains or an 8bit microcontroller. Singularity means only to build machines who are able to play games, especially complicated language based games. That means, there is no need to determine the brain functions on an atomic layer to understand Artificial intelligence. Most programmers of Game AI didn't know the difference between a transistor and a resistor and their Artificial Intelligence software is able to beat any human player in chess. So we can argue, that intelligence isn't located in the brain but in the games humans are play. Games are social constructs, they were invented in a certain cultural background. A racecar game has another ideology then a capitalism game in which the goal is to earn as much money as possible.

It is not possible to locate in a racecar game any kind of biological motivated DNA. And it makes no sense to analyze the DNA of the drivers to measure their IQ. To understand a specific game we must describe the art history of the issue. For example at which time it was played first, which types of cars are used and so on. That isn't a task for biology, but a task for ludology which is according to WIkipedia in the domain of Media studies and Humanities, that means it is located outside of Mathematics and outside of science. Intelligence research is not a subject for scientists but for historians, art history experts and cultural studies authors.

This seems to me an extremely confused question.

Research overturned the old belief that IQ is a mater of pedigree. The current perspective is that from 6 months old to adolescence, nurture dominates over genetics.

Research on the contrary shows, that in developed countries IQ is highly heritable, further, that the effect of nurture aka shared environment is extremely small, and also, that all gains that may be made with early childhood interventions are temporary at best.

It is at least interesting that multiple numbers are used to gauge academic achievement yet all the complex genetic and educational factors result in a one-dimensional number, IQ, that scores a person's intelligence.

The justification for the "one-dimensional number" is the fact that the test results from all these different tests are highly correlated. That makes it possible to extract a common factor, the g-factor, from a battery of cognitive tests. This g-factor is the predictive part of IQ in life success and educational attainment, and also the heritable part. A good IQ test extracts a g-factor from a multitude of tests.

Even if there are no further intelligence genes found, twenty-two genetically distinct, independent degrees of freedom define the variable features of human intelligence. Thus, a scalar value cannot possibly quantify the permutations of genes that human intelligence. At least a 22-dimension vector is required to represent the intelligence feature set.

There are likely thousands of single nucleotide polymorphisms that contribute to the variation in human intelligence. The relevant thing in real life is still the g-factor. A car may be described by a very high dimensional blueprint. Its top speed is still just a single number.

"We buy IQ" because it is highly predictive of life outcomes, especially educational attainment. IQ research is the biggest success story in psychology/social sciences. The existence of the g-factor validates the concept of IQ.

• What is this g-factor you are mentioning? – Răzvan Flavius Panda Oct 17 '18 at 14:49