Can we see where math begins and science ends?

Galileo is often called the father of modern science because of an insight he had about the relationship between mathematics, and what we are able to see in our world. Two of John Horgan’s recent blog posts (and the writing to which they refer) nicely demonstrate what I think is a remarkable oversight in discussions about the prospects for the future of science as we know it.  Neither John, nor any of the writers to whom he refers, consider the significance of the role that mathematics is playing in the development of scientific ideas and analyses.  None of them wonder about how mathematics shapes our views of reality.  If we want to consider that we’ve reached some limit to the progress that science can make, perhaps we should revisit Galileo’s original insight about what science is, and think again about the role mathematics plays.

Galileo understood that science could not be done without mathematics.  And it’s this science that so many seem to be worried about. From his book Il Saggiatore (The Assayer) published in 1623:

Philosophy [i.e. physics] is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.

The words that strike me are “continually open to our gaze, but not understood.”  Galileo’s insight was that mathematics could bridge the rift between the gaze and the understanding.  But how or even why does it happen?  Trying to get at the how might shed new light on what we call physical law.  Getting at why might lead to fresh ways to consider questions about consciousness and objectivity. These questions are at the bottom of any other questions we might have about the value or future of science. I’m not meaning to suggest that there are any easy or definitive answers to these questions, but they are certainly relevant to the questions being asked about what science has or may yet accomplish, and consistently overlooked.  The Euclidean geometry that lit Galileo’s way was not originally motivated by pragmatic concerns or by the use Galileo made of them.  There is something independent about the spirit of mathematics that we barely understand.  And the mathematical landscape has exploded with ideas since Euclid’s survey of geometry.  It is the wealth of conceptual forms, provided by mathematics, that has shaped the often bewildering, counter-intuitive ideas in modern physics.  It is mathematics that resolves the flood of experimental observations into the space-time continuum of Relativity, or into the quirky laws of quantum mechanics.  It is mathematics that leads some physicists to consider multiverse ideas, or leads others to the dispute over which is primary to the universe – material or information.  It should not be possible to leave mathematics out of a discussion of science, and of physics in particular.

Questions that address the emergence of mathematics, and the cognitive structures it may mirror, could give us a new way to tackle the enigmatic relationship between mind and matter (as we now imagine them).  Mathematics is, after all, an almost purely introspective science, yet it builds the science of material, the structure of modern physics.

I remember reading John Horgan’s 1996 book The End of Science and really enjoying it. The intimacy of his interviews brought real vitality to the ideas.  His latest post is called The End-of-Science Bandwagon is Getting Crowded.  In it he quotes from some of the responses to question, What should we be worried about?  He’s chosen scientists concerned with the future of science.  Horgan also references a Nature essay by Dean Keith Simonton who argues:

Our theories and instruments now probe the earliest seconds and farthest reaches of the Universe, and we can investigate the tiniest of life forms and the shortest-lived of subatomic particles. It is difficult to imagine that scientists have overlooked some phenomenon worthy of its own discipline alongside astronomy, physics, chemistry and biology…Future advances are likely to build on what is already known rather than alter the foundations of knowledge.

But it seems to me that if we can manage to get our attention on some useful questions about the more specific nature of this knowledge – about what it’s made from and how we built it – then we might begin to see a revolutionary way to extend its limits.  We might replace our current notion of objectivity with something more appropriate to the interplay between ourselves and the world that we see in pure sensation as well as mathematics.  Mathematics is not just the tool.  It is the strategy.



3 comments to Can we see where math begins and science ends?

  • happyseaurchin

    Thanks Joselle. Hopefully one day we shall have something more of an engagement than my rather brief and enigmatic commentaries.

    Can’t agree with you more. And your ability to put words to it is quite marvellous, and well beyond mine. I certainly appreciate your observations, explorations and insights.

    If we are to recognise that mathematics is an action in which our consciousness participates, this suggest — at least to me — a present-minded process. We can adopt the current scientific methodology which is to place it as the object of attention, or we can adopt a different methodology that allows us to become aware of these processes as they occur. That is, increase our sensitivity to these processes. My intuition suggests buddhism and its radical subjectivism, coupled with thought-experiments.

    That is, I do not mind going over the weaknesses of the scientific methodology, its blind-spots, but this is still within the reference field of science. The jump required is to actually consider an alternative methodology, and just the results as they arise, and not in direct comparison to science. The adoption of an alternative methodology (for specific investigations and subject matter), entails the adoption of different means of judging results, or rather evaluating efficacy.

    I look forward to a time when we feel confortable enough to make this jump, even if it as temporary as an extended thought-experiment.

  • happyseaurchin

    Thoughtful thread through this, and i know i bring my own agenda to this while reading — I keep thinking you are going to state something that rings completely true for me, but you always skirt the very edge of it and never jump.

    I was just thinking about vectors last night while my mind settled (or in this case not) to sleep: how obvious the vector of conscious attention is pointed outwards, ever outwards at objects out there in the universe, and such little attention is paid to reflect on the aspect of the vector that come from within. We are both interested in this.

    To me, however, it requires a different methodology if we are to make significant progress — it is simply not enough to point the vector of attention inwards.

    You ask the right questions but you never seem relinquish the methodology that has evolved for the examination of objects, aka physics, and thus keep putting mathematics as the object of inquiry

    I must apologise because I am fairly certain I have been giving the same response over the years and surely you must be tired of it. I am. But you are in a position of responsibility, authority even. You are on the inside of academia, and I am on the outside. I have ventured in on numerous occasions, and your posts here I enjoy, thank you. Perhaps one day I shall provide such an invitation that you can not refuse.

    In relation to Horgan, there is a massive phenomenon that scientists have overlooked, as you address readily. It is not another discipline which fits nicely along side others. It is simply that the methodology that scientists use is a specialist methodology, and another is required to… intuit and reveal and bring to our awareness, the consequence of which is not another set of philosophical platitudes, but insight and direct application to many subjects, perhaps especially artificial intelligence, certainly our social sciences, and impact in our major institutions, education amongst others.

    • Joselle

      I understand what you say, although I’m not sure I understand fully your own ideas about mathematics. But I would like to clarify one thing about my approach to all of this. What may look like my making mathematics the ‘object of inquiry’ is just a first step. If it’s possible to change the way we think about mathematics (or don’t think about it!), it may be that mathematics can become understood as an action (in which our conscious will participates, but has not actually designed) and this can open many doors (to the nature of perception, the role of empiricism, what it means to know something). This would inevitably facilitate asking the right questions about what’s lacking in many of our current methodologies.

      I do appreciate this discussion (and don’t tire of it).