Notes About Continuity and Discontinuities


This evening, I am giving a brief talk to a Unitarian [1] humanist group on discontinuities. “Discontinuities” is the title of my upcoming book, and the title of one of its chapters. The talk will be followed by a discussion.


This document is rather sketchy (not thorough or detailed). It presents some of my personal notes for this talk. Some of it is text I will recite verbatim this evening. Some of it is written to you, the reader. The post is not a summary of my book, nor of its chapter on discontinuities. So if this topic does not interest you, please don’t be put off from delving my upcoming book! [2]

The abstract of my talk is:

It is often said that the sophisticated alternative to binary thinking is continuous (shades of grey) thinking. I will argue that possible and actual worlds and mind involve multiple discontinuities.

Credit to Aaron Sloman

I was introduced to the epistemological and scientific importance of continuity vs. discontinuity by Aaron Sloman. He has written more on this subject than anyone else. It is due to reading and discussing this subject with Aaron from 1990 onwards that I have pursued this matter.

However, one should not blame Aaron Sloman for any misunderstandings of mine. Best to read his sources directly. See the Bibliography below which while incomplete has pointers to some of his papers and web pages on the subject.

See also some personal history below.

Relevance of this topic

As you listen to this talk you’re likely to wonder “Why on earth does any of this matter?” And perhaps “Why did I come here tonight?”

The answer is that the benefits of recognizing, challenging, and correcting flawed fundamental assumptions about fundamental aspects of physical, biological, social and mental world are largely unpredictable. But the more fundamental the assumptions, the larger the potential impact. And here this evening, we are talking about very foundational matters. If nothing else this might help you spot errors in your own theories and discourse and in those of others. It might also lead you to new insights, such as about other things to do on a Sunday evening.

Bruner assumed continuity is the modern and rational assumption of science

In a 1956 paper discussing Freud’s contribution to our conception of humans, the great cognitive psychologist, Jerome Bruner expressed what is now a widespread belief amongst intellectuals:

[ancient] Greek physicist-philosophers […] formulated a bold conception of the physical world as a unitary material phenomenon. […] Since that time, the phenomena of the physical world have been conceived as continuous and monistic, as governed by the common laws of matter. The view was a bold one, bold in the sense of running counter to the immediate testimony of the senses. It has served as an axiomatic basis of physics for more than two millennia. The bold view eventually became the obvious view, and it gave shape to our common understanding of the physical world.

Basically, Bruner is saying that the physical world is a continuous world, and that science has both assumed and proven this.

Bruner then goes on to praise Darwin for postulating that humans are the product of a continuous process of evolution. On Bruner’s interpretation of Darwinian evolution, whatever qualitative differences humans may evince compared to currently living animals, or the fossil record, are merely superficial differences that ultimately can be explained by an underlying quantitative process that varies continuously (rather than discretely).

He then praises the great psychologist, Sigmund Freud[3], for bringing what he takes to be the wondrous, powerful and necessary assumption of continuity to psychology.

There are many realms to which concepts of continuity and discontinuity can be applied. Similarly, there are many types of actual or possible continuity and/or discontinuity. Like many people who broach this rarified matter, Bruner conflates two (and perhaps more) ontological realms, namely transformational / temporal continuity and structural continuity of the world as objects of study. There is another related set of distinctions (types of continuity or discontinuity) that Bruner assumes and in relation to which he takes position: namely quantitative ways of thinking about (or modeling) scientific subjects. He believes continuous quantitative ways of thinking about scientific matters are ultimately superior to discrete ways.

The thesis, rephrased

Physicists have been extremely successful at describing and predicting events using continuous quantitative mathematics. The success of quantitative methods in physics and chemistry led to a conjecture and eventually a belief that the same approaches could and should be used in cognitive science, i.e., to understand the animal and artificial minds. In fact, quantitative approaches have been increasingly popular in artificial intelligence and other realms of cognitive science; and there is no denying they already have had considerable success. However, and unfortunately, success of continuous quantitative approaches in physical science, AI and cognitive science has has reinforced the false belief that ultimate reality, and all reality, can ultimately be most correctly characterized in mathematical terms. An extreme version of this, on which we will not dwell today is the idea that the universe itself is mathematical. In any case, in all of the above, when they push speaking/thinking in “mathematical terms”, academics typically mean continuously varying quantitative mathematical modeling. It is easy to forget discrete mathematics and many other forms of mathematics (and other realms of knowledge) that are discrete, not continuous.

While I certainly agree that quantitative methods, and hence being educated in quantitative mathematical theory and techniques, are essential to understanding reality, from the physical level to the highest levels, I will argue that at all levels one encounters discretely varying phenomena, which cannot adequately be reduced to continuously varying mathematical models.

Continuity vs. discontinuity

In this part of the talk, I will say a few words about continuity/discontinuity per se. I’ll use the white board.

In sum, to my knowledge (and I am not a mathematician) the domain of real numbers is the fundamental mathematical basis for continuity. Many domains in mathematics deal with the domain of real numbers, such as topology.

There are also branches of mathematics that are not (inherently) concerned with “the continuous”, such as discrete mathematics.

The importance/necessity of quantitative mathematics

Just to reduce the odds that I be misunderstood, I will literally reprise what I wrote above

While I certainly agree that quantitative methods, and hence being educated in quantitative mathematical theory and techniques, are essential to understanding reality, from the physical level to the highest levels

None of what I am saying ought to be construed as advocating against continuous quantitative mathematical literacy. Many kinds of quantitative mathematical literacy are necessary for rationality, science and society. It is important to be trained in several of

  • Mathematical analysis,
  • quantitative algebra (matrix algebra),
  • differential calculus,
  • probability, and
  • Bayes

See Mathematics Subject Classification scheme in Wikipedia.

Lack of quantitative mathematical literacy is the source of many of our deepest problems. We need quantitative mathematical literacy. But we also need literacy in discrete mathematics, logic and philosophy and science.

Examples outside of art

I will discuss some of these examples:

  • classical music: the notation and performance of classical music is digital (e.g., pressing keys on a keyboard, opening and closing holes on flute, holding keys up/down). Yes, there are continuous aspects to this, and it’s an important part of this thesis.
  • Contrary to Bruner’s interpretation, there are key qualitative assumptions in Darwinian evolutionary theory
    *DNA and genes are actually great examples of discrete information! (Bruner’s article appeared shortly after the discovery of DNA. Maybe he would have revised his thesis if he had written his article later.)
  • understanding (whether schooled or unschooled) geometry involves discrete knowledge. ( Knowledge is discrete).

Information layering: stacks

One of the most important human discoveries is that information processing systems can be constructed from discretely varying mechanisms. The Turing Machine is the basis of modern artificial computing. Almost all modern computing is based on digital (discrete) structures. And almost all autonomous systems involve discrete forms of information, even if designers try only to use quantitative concepts to describe them.

The Internet Protocol (TCP/IP) and similar communication protocols also involve an information stack: the protocol stack. It defines several discrete layers, which each internally admit of discrete variability. The application layer often has several layers as well. Software developers create and implement all kinds of ontologies (discrete objects and processes).

Evolution reinvented this.

Here’s a related (quite compact) footnote from my first book:

Even within a single modality there are many layers of processing and representation. In the case of vision, for example, the detection of lines and edges requires several layers of grouping.  Unfortunately, these discontinuities (and multitudinous layers) are often lost in computer models, including some connectionist models, as an artifact of the mathematical tools used by scientists. Claude Lamontagne  formalized the Gestalt notion of grouping (using a principle of adjacency). He and I applied it to the evolution of fine-grained perceptual processes from the retina upwards  (Beaudoin, 1990; Lamontagne, 1987). In our theory, each layer of grouping is an implicit rerepresentation of a lower-layer. Strictly speaking, informational layering is discrete, not continuous, though it allows the system to construct fine-grained representations that are treated as if they were continuous.

Binary thinking ain’t always so bad

Manicheanism has gotten a bad rap. Contrast

Examples in art

Given my project on Learning from Stories and Other forms of Art you should expect me to provide some examples from art, and to ask you to do the same.

It may be helpful to consider discontinuity by contrast with attempts to remove it:

The Discontinuities: Love, Art, Mind book will feature

  1. a painting that illustrates blending of continuous and discontinuous transitions. Stay tuned.
  2. a series of diptychs by Lam Wong accompanying a short story: letter-painting pairs.

Popper’s 3 worlds — augmented

I don’t think we can adequately take a stance with respect to questions about discontinuity and continuity without explicitly or implicitly considering the distinctions in Popper’s 3-world cosmology. That’s because , whatever one’s stance, one needs to say how it applies to each of these three worlds and their derivatives. (The latter qualification implies the 3-world cosmology is incomplete. … Has anyone previously asked how incompleteness theorem applies to it?)

In Cognitive Productivity: Using Knowledge to Become Profoundly Effective, I discussed and tried to enhance Popper’s 3-world cosmology. See Popper’s three worlds – Wikipedia:

  • World 1: the world of physical objects and events, including biological entities
  • World 2: the world of mental processes, and
  • World 3: objective knowledge

I replaced Popper’s World 2 with something more general, which I called World 2′ (“world 2 prime”, or “World 2 VM”), i.e., the world of virtual machines.

World 3 more generally is the world of artefacts. See Carl Bereiter’s 2002 book Education and Mind in the Knowledge Age. Objective knowledge is the most interesting subset of World 3.

Each of these worlds itself admits of quantitative and qualitative variety. Objective knowledge, however, is essentially discrete. Even knowledge about continuity is itself discrete.

The best taxonomy of knowledge is THE COMPUTER REVOLUTION IN PHILOSOPHY (1978): Chapter 2.

In his 2002 book, Carl Bereiter characterizes understanding as relations between knowers and objects of knowledge. I discuss this in Cognitive Productivity: Using Knowledge to Become Profoundly Effective. Note that relations are also discrete.


I am not a mathematician, physicist nor a geneticist. Those are key disciplines for this topic.

Bibliography (VERY incomplete)

Here are some hastily gathered references pertinent to our topic. I may later add further links to this post.

Seminal and ongoing work by Aaron Sloman

As I mentioned above, no one has done more than Aaron Sloman to demonstrate the deep relevance of discontinuity. While this is in effect philosophy, it draws from and has deep implications for all science. (The distinctions between empirical science, philosophy and mathematics are not as clear cut as people are led to believe by most scientists, philosophers and mathematicians!)

Merit is not a one-dimensional scale, but may be a complex and perhaps even inconsistent partial ordering implicitly defined by a large and not necessarily consistent collection of motive comparators, built up through a long and erratic process of learning (phylogenetic and ontogenetic). Thus theories of decision making based on scalar representations of strength of motives may be inaccurate and unable to account for some of the richness of human and animal behaviour.

Mental mechanisms cannot be completely ordered in respect of degree or amount of intelligence. System A may be more intelligent than system B in some respects, less in others. For instance, it might be argued that an ant colony forms a system which is in some respects more intelligent than a human being, since more different concurrent processes can be devoted to a common set of goals, though in other respects it is less intelligent, since it does not produce intelligent behaviour derived by reasoning about an explicit set of goals and facts (as far as we know).

There are many different mental states and processes which have features in common with emotions. The boundaries between the different concepts are not very sharp, but some broad distinctions can be made

The space of possible intelligent systems is not a continuum: there are many discontinuities giving it a rich discrete, non-linear, structure. A full survey would need to explore many sorts of discontinuities. A study of different kinds of animals and their abilities (e.g. Lorenz 1977) would help to draw attention to important subspaces. Equally, an overview of different subspaces should help to organise research on animal capacities, and the evolution of intelligence. Some fairly obvious discontinuities include the differences between systems which do and systems which do not have the following abilities:

Also much of what intelligent agents are trying to achieve, avoid, maintain, is not concerned with physical states that they can directly and continuously control. Rather it is often concerned with the distant future, absent objects, events that might occur but haven’t yet. This means that the control has to go via “representations” of those things, i.e. information structures with semantic relationships to other things.

Investigation of control systems with these two features (partly) non-quantitative, and (mostly?) semantic require concepts and techniques that so far have been developed within the study of information processing rather than the study of electronics, chemistry, physics, (traditional) psychology, physiology, etc.


Perhaps the most striking challenge to the predominant view that biological development is largely dictated by smooth, long-range biochemical gradients comes from recent evidence that digit formation in the embryo can be regarded as arising from striped Turing patterns [39]. Here too, Wnt proteins play a role. Digit formation is ultimately under the control of a gene called SOX9, which triggers differentiation of soft tissue towards the formation of bone and cartilage. WNT gene products and bone morphogenetic proteins (products of BMP genes) influence the activity of SOX9 in a manner described by an activator – inhibitor scheme, so that drug-induced suppression of WNT or BMP leads to predictable changes in the number or spacing of digits.

  • E. Tory Higgins has been arguing against utility theory (as I did in my Ph.D. thesis in 1994).

Here is Keynes, the father of modern economics, making a profound statement about human motivation that other economists-and other social scientists-have not taken seriously enough. Most social scientists continue to take for granted that it is expected utility-“the outcome of a Weighted average of quantitative benefits multiplied by quantitative probabilities”- that defines what people really want and underlies the preferences revealed in their choices. What about the alternative that Keynes is identifying? What about “animal spirits”? What about the “spontaneous urge to action rather than inaction”? Keynes is saying that this urge is critical to motivation.

Some of my work

Views contrary to the above

Views contrary to the above abound. They are the mainstream. To take but a few examples (very important works).

  • Merica 2004 “State transitions between wake and sleep and within the ultradian cycle with focus on the link to neuronal activity”. This is one of several papers that makes the rather obvious point that sleep stages (N1, n2, n3, rem) is an arbitrary classification. Borbély had previously made the same point, arguing for continuity.
  • The Cognitive-Emotional Brain: From Interactions to Integration by Luiz Pessoa. Argues that mental resources and brain/information processing is continuous.

Footnotes (not necessarily in order of appearance in the text)

Footnote 1

Unitarianism does not have any dogma, just shared values and principles. Our humanist group consists of atheists, agnostics and a couple of anti-theists. We do not discuss “religion” and “god”. In fact, most of us consider the “god” meme to be at best irrelevant but in fact a backward, harmful one.

Unitarianism, while often called a religion, in Canada is not truly a religion in the traditional sense, nor is it a “church”. When Unitarians call Unitarianism a religion and their meeting place a “church”, it is normally because they have failed at conceptual analysis, and are unconsciously perpetuating a false tradition.

Footnote 2

The Discontinuities book only addresses the (admittedly abstruse) topic of discontinuities head on in one chapter. In (and indeed between) other chapters, it’s an underlying theme that many readers can simply ignore. You can think of the book as a very deep chocolate cake. You don’t have to try to dig your fork all the way to the bottom. There are many layers that you can ignore.

Footnote 3 (Re: Sigmund Freud)

Not everyone agrees that it is correct to call Freud a psychologist.

Personal History

I mentioned above that I was introduced to the importance of discontinuities by Aaron Sloman. Given that so much of my work is an offshoot of Aaron’s projects, I thought I should provide some further background. I write specifically about Aaron’s work in my future book, Great Contemporaries in Cognitive Science and Related Endeavours. See also Aaron Sloman: A Bright Tile in AI’s Mosaic

As an undergraduate, before meeting Aaron Sloman, I took several courses in logic, epistemology, other realms of philosophy, discrete mathematics, linguistics, and perception. I was introduced to Immanuel Kant’s work by Claude Lamontagne who taught the perception course I took. (Lamontagne would later win provincial and national teaching awards for that course.) My epistemology prof might also have mentioned Kant — I can’t recall. In any event, I became quite interested in the implications of Kant’s views for understanding human perception, causal understanding and philosophy of science. As an undergraduate, I wrote a paper that challenged Humean assumptions in psychology. And my honour’s thesis was essentially Kantian.

When I was an undergraduate, Claude Lamontagne told me about Aaron and recommended I consider Aaron Sloman as a Ph.D. thesis supervisor. I did not read Aaron Sloman’s work until after my undergraduate degree. However, I based on what Claude told me, I was confident the fit would be excellent. I applied to do a Ph.D. thesis with Aaron on a Kantian/AI modeling of human causal reasoning. In retrospect, I can see why Aaron might have appreciated receiving my application.

Aaron was at Sussex University when I started my D.Phil with him. He moved to the University Birmingham, England, in 1991 and has been there since. I followed him there (and epistemically). Aaron was then launching/rebooting his Cognition and Affect. I decided to suspend my work on causal reasoning and jump onto his project. I would soon realize that Kantian ideas were applicable to affect.

I’ve long intended to return to geometry as well. But that is for the future.