Proofs and Prompts

Notes from the common room

The Platonist view

Uri Bader, Professor at Maryland University and Weizmann Institute

Math exists. We explore it.
We do not invent math. We invent ways of understanding and communicating it. We choose which parts of it to study, and we develop an adequate language for that that suits us, humans. But we explore an objectively existing domain. Empirical fact: our abstract exploration bears fruits also in the physical world. This fact is often used as a retrospect justification, but it is rarely our initial motivation. A common metaphor for math exploration is landscape charting.

Geography exists.
Past cartographers sailed the seas. There was an heroic aspect to their profession. There was an artistic aspect to map drawing. Could they envision satellites and computerized general information systems? Their profession is less romantic now, but our maps are better.

Can we envision how math will be explored in years to come? I believe that we will ever be stunned by its inner beauty and astonished by its interplay with the physical world. This interplay will remain fruitful and relevant to society, perhaps more than ever before, and so also the math community. Yes, we are facing a challenge and we should publicly discuss how to adjust. However, we better stop ranting about progress, rather openly embrace it. We should celebrate new opportunities to better fulfil our mission.

Received 21 August 2026.

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9 responses to “The Platonist view”

  1. Morten Krogh Avatar
    Morten Krogh

    Yes, exactly. This is the only reasonable way forward. All knowledge work is changing dramatically. That can be scary, but also exciting. Embracing the new world is the most sensible option.

    Reply
  2. Uri Bader Avatar
    Uri Bader

    I sent this text over three weeks ago, and a lot has happened before its publication today. My aim in writing it was to share my optimism and excitement with some of my colleagues and students who seem desperately pessimistic. I firmly stand behind it.
    However, I wish to clarify that I am not blinded by this optimism. I am concerned with the development of unsafe technology and with the economic effect of irresponsible adoption, which are enhanced by delegating unmonitored power to private, typically predatorial, entities. These concerns are not particular to math, and we should all face them. I am also concerned with some possible effect on the math culture and community. We should face these too. In fact, we have started to do so and I am optimistic regarding our capability. In the text I deliberately don’t touch the question of how to confront these challenges. Rather, I am looking to the distant future, excited, and I am inviting the reader to join me.

    Reply
    1. Galyna Livshyts Avatar
      Galyna Livshyts

      Exactly!! I think one needs to distinguish between two issues: 1) the way the AI influences mathematics and the process behind it and 2) the fact that AI is a commercial asset, and reliance on it undermines the free access to knowledge and equal opportunity.

      Basically, I think that 1) — embrace (like you say), and 2) — construct an open-source AI universally used by all mathematicians in all the universities, starting with undergraduate students (like mathscinet it might be special to math), train it on math, make it better than the commercial models!

      I also have a post about it waiting the 3 week backlog:) I might modify it by the time the turn gets to it. Assuming the humanity is still in place in 3 weeks:)

      Reply
  3. Maksym Zubkov Avatar
    Maksym Zubkov

    “But we explore an objectively existing domain. Empirical fact: our abstract exploration bears fruits also in the physical world.”
    Are we sure that math is not just a mass hallucination and artifact of our brains as a response to millions years of evolution of learning and constructing algorithms of how to reproduce the world model within our neurons?
    It’s a genuine question that I ask myself from time to time. I just think that we will need always challenge our foundational knowledge and stigma. In the same way, (un)fortunately how tech progress and ai companies challenge our math world right now

    Reply
    1. Uri Bader Avatar
      Uri Bader

      Sure? Surely nothing is sure!
      Everyone knows that no one knows anything for sure.
      I think that I think therefore I think that I exist, but verbal thinking might be nothing but text manipulation. Let’s not go there. Let’s discuss math, or at least what we think of it. Better yet, let’s discuss poetry. Sometimes I read a line so wise that I know that I couldn’t come up with it. There is someone else out there. Math, likewise, is wiser than us. You could play with it formally, you can squeeze it like a piece of dough and get the illusion that you freely form it with your imagination, but then you stumble onto something that was always there, say the classification of simple algebraic groups (yes, I am biased), and you know that no human had formed that shape. I can go on and try to better make my case, but I suppose this is not the right forum for this discussion, so let me conclude with the following unbeatable argument.

      Reply
  4. Andreas Thom Avatar
    Andreas Thom

    It seems that AI might turn the platonic view (which I do not share) into a solid reality. Math exists and AI can provide answers to most questions we can think of without us understanding what is going on. The lack is then only our understanding of it and not the answer per se.

    Reply
  5. Marcelo Muniz Alves Avatar
    Marcelo Muniz Alves

    This text might be a little too optmistic, but to me is the first idea that makes some sense on this “brave new world” where we’re being shoved in by AI. The feeling is akin to that of the period of the great explorations although, in our case, there’s no end of discoveries still awaiting. It is impossible for me to conjure the faintest idea of what will be this new geographic study of the mathematical landscape, but this text gives me some hope that it might not be as bad as we fear now.

    Reply
  6. exactlyec626f75e3 Avatar
    exactlyec626f75e3

    How about we having AI help explore the whole Platonic universe of Math for us and also elaborating on its discoveries to contribute to our understanding of them?

    Reply
    1. Uri Bader Avatar
      Uri Bader

      How about having a hiking robot? Will that make you stop hiking? On the contrary, you will probably use it to explore new paths. If a machine could love, will it stop you from loving? Maybe I do not understand the intended question here.

      Reply

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