Riffing Deacon, Dehaene, Rotman, and Tegmark: More on the Nature of Mathematics

51VY4T00VCL._SY344_BO1,204,203,200_Is mathematics an invention of the human mind, or is it a universal “eternal truth” that we humans discover?

I once wrote a blog post about Brian Rotman’s book, Ad Infinitum, in which he advocates “taking God out of mathematics and putting the body back in”. 510+TGtLoML._SX258_BO1,204,203,200_

Since writing that blog post, I have thought a lot more about the reality of math. And one reason is that I have since written a book about a set of fractal curves that I “discovered”. Or, did I “design” them? (read on to find the answer :)

fractals

Brain Rotman recently wrote a review he wrote for the Guardian on the book Our Mathematical Universe, by Max Tegmark.

Rotman’s review is quite critical. Tegmark claims that mathematics is the very foundation of the structure of the universe. He goes as far as to say that “Our reality isn’t just described by mathematics – it is mathematics”.

511tVkk9X5L._SX258_PJlook-inside-v2,TopRight,1,0_SH20_BO1,204,203,200_Rotman would be one of a growing number of mathematician/philosophers who take issue with this kind of claim. They would counter that mathematics is a human invention. My first encounter of this idea was through Lakoff an Nunez in their book, Where Mathematics Comes From – How the Embodied Mind Brings Mathematics into Being. This book explores mathematics from the point of view of cognition and linguistics.

All very well and good. Math comes from human brains. But the more I engage in mathematical activities, the more sympathetic I become with mathematicians who believe that they are “uncovering” great truths rather than creating ideas out of thin air. While exploring mathematical ideas, the feeling that I am discovering something outside of myself is very strong. What is the reason for this feeling? What is happening in my brain that makes this feeling so strong?

Why I am Becoming a Mathematical Agnostic

I am just about finished reading The Number Sense by Stanislas Dehaene, who is working on the cutting edge of brain imaging and identifying cerebral structures associated with abstract number and calculation. 

Dehaene believes that math is a human invention.

However, he acknowledges the long reality that precedes us. According to Dehaene, the notion of number is perhaps the most primal mathematical construct that our brains perceive, and we share a rudimentary number sense with other animals.

TurnerPremaNo surprise here: the universe – at least on the scale of mammals and birds, has clumps of matter and events that we perceive and that are meaningful to us. And of course the biosphere is doing its part to continue clumpifying matter and events, including me writing this blog post.

“Number” could be seen as an emergent property of the biosphere. Animals have evolved behaviors associated with number (as well as symmetry, and other attributes associated with math). In turn they impose their emergent representations back onto the environment as part of an ongoing feedback loop of complexification. Humans have simply taken this feedback loop to a conscious level.

A few years back I read Incomplete Nature, How Mind Emerged from Matter by Terrence Deacon. Although his writing is painful, the ideas in the book are very thought-provoking. What I learned from Deacon makes me pause before heading straight to mathematical atheism. I recommend this book. Deacon points out the many levels of emergence that forged the structures of our universe, and ultimately, our minds.

My conclusion: to say that math is strictly a human invention is taking it too far. Math is not arbitrary. We are made out of structured matter that exists throughout the universe. Our bodies, brains, and minds are tuned-in to that structure. We evolved with it and in it. Our language, as well as our genes, are “about” the biosphere, which is “about” the universe.

On Discovering vs. Designing

In Brainfilling Curves, I chide Mandelbrot for saying he “designed” certain fractal curves. I can’t say that I blame him, considering how clever the snowflake sweep is, for instance. But I prefer to say he discovered it.

Screen Shot 2014-02-01 at 8.14.45 PM

And I claim that many of the fractal curves in my book were discovered by me. But it gets a little fuzzy at times. Geometrical objects range in primality from the circle (which no one would claim to have designed) to rare, high-order self-avoiding space-filling curves, many of which I have spent years uncovering. The more rare, the more unlikely the geometrical object, and the more information required to describe it, the more like design it becomes. I see it as a continuum. At the far extreme of this continuum would be a painting by Kandinsky.

colourful-ensemble-1938

I believe that we feel the sensation of mathematical discovery because of the evolution of our brains, language, and culture, from which we cannot escape.

The evolution of our brains, language, and culture are a continuation of the evolution of the structure of the universe. Thus math, while it may not originate from the universe, is a language we invented which is finely tuned to it.

And since we are made of the stuff that our math describes, our math feels…

perfect.

 

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8 thoughts on “Riffing Deacon, Dehaene, Rotman, and Tegmark: More on the Nature of Mathematics

  1. NASA uses prime numbers to communicate with potential alien intelligences because math is universal. If you expect that given enough time, everyone will “invent” the same thing – it seems like you’re describing “discover” not “invent”. And honestly, claiming that “invention” is somehow different or superior than “discovery” feels like hubris. Man did not create fire, nor math.

    • Hi Qarl,
      thanks for the interesting comment!

      NASA’s theory is that math is universal – and this is why they use prime numbers. I think it is extremely likely that the theory is correct.

      But it’s a theory. We have not yet had any response from extraterrestrials.

      I do not expect that given enough time, everyone will “invent” the same thing. The implication I wanted to get across in my post is that the likelihood of two people inventing/discovering the circle is much greater than the likelihood of two people inventing/discovering the Gosper curve, which is greater than two people inventing/discovering the Kandinsky painting shown in the illustration.

      The likelihood that an exact replica of that Kandinsky painting appearing sometime somewhere in the universe is very very close to zero. But it’s not zero.

      I am intentionally blurring the distinction between discover and invent. It’s a mental experiment I’m conducting to encourage readers to get out of their dogma houses.

      Also, I did not claim that invention is different or superior to discovery. Perhaps I did not express myself well.

      Also, I think it’s an error to put “fire” and “math” in the same category for comparison. Fire exists as a physical phenomenon in the universe. Mathematics, on the other hand, is “an abstract study” that “seeks out patterns” in the observable world. It is not a thing, it is not stuff, it is not a form of energy. By implication, math is a representation created by an observer or some entity that finds patterns. And by “observer” I include the biosphere in general, which is where I believe math originated. This is compatible with the relatively new field of biosemiotics.

      I’m just interested in getting people to come out of their dogma houses – on both side of the debate.
      -j

      • oh no! i had asked your site to email should anyone respond – but it never did. i just happened to come back here and see your response.

        there are many places i could respond – but i think the most interesting one is this: i disagree with your intuition regarding fire and math. i think it’s much more likely the underwater creatures of europa will discover prime numbers before they discover fire.

        as you say, math is the study of patterns. which is in fact the study of information. information is DEFINITELY a physical feature of our universe.

        K.

  2. Ah…interesting point there Qarl … that the underwater creatures of Europa will discover prime numbers before they discover fire.

    Your final comment – that information is a physical feature of the universe – is really interesting. The more I learn about biosemiotics, the more I see that life itself is a dance of information over matter. In fact, I like what people like Chris Langton, Terrance Deacon, and others say: the phenomenon of life can be seen as a shifting of control away from matter and energy towards information processing (adaptation, biosemiotics, genetics, intelligence, etc.). This reverses the universal trend towards entropy (at least locally).

    Taking the idea all the way down the structure of the universe in general is interesting.

    Whether this means that the universe is made of “math” or not remains a debate for me. I see “math” as coming from observers of things, not as an innate property of things.

    But to consider that the universe is made of information…that makes your argument a bit more compelling.
    -j

    • “But to consider that the universe is made of information…that makes your argument a bit more compelling.”

      i believe that’s just a rephrasing of the entire topic here: the universe IS math. that’s the jist of the claim.

      you might be interested to know that like energy – the universe conserves information. this tidbit is the snowball which started the avalanche of universe=math.

      best,

      K.

  3. Tegmark is being a shrewd marketer for his book. He is stoking the fire of the debate by making the claim that the universe IS math.

    I don’t think one man and one book will be able to change the definition of the word so easily. Math is the language of Science. Humans invented Science, and they invented math to describe it.

    The real debate is whether the patterns that we see and describe with math are universal or arbitrary. Tegmark’s abuse of language (for the sake of promoting his book) has confused the debate.
    -j

  4. Hey Jeffrey. Saw your post about Slow Programming on Hacker news and stumbled upon this post also. Flipped through your book on Brainfilling curves, really liked the nice illustrations there.

    Have you created some software to interact with this kind of geometry realtime ?

    I’ve been working for the past 3 years on http://GeoKone.NET, I think you might like it. Check out the introduction video on the front page :) It’s completely free to use and runs in your browser, using HTML5/Javascript. With GeoKone you can generate recursive, natural geometry.

  5. The Evolution of Mathematics on Planet Earth – Nature...Brain...Language...Technology...Design

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