I started this blog in 2014 after I retired as a place to put ideas of many sorts online. At the same time, because I had papers in multiple specialities, I decided to make an online archive too, an idea I heard first from Martin Groetschel when digitization had just begun to be used widely. I set up this personal website at Brown University to do both. This archive is still live, accessible wby clicking the other items in the "navbar" at the top of the page.
In 2021 I approached the American Mathematical Society to see if they were interested in collecting expanded versions of some of my posts as a book. I worked at that time with their editor Eriko Hironaka. Although there was some disagreement, I shortened controversial parts and Numbers and the World: Essays on Math and Beyond was published in 2023. But I continued to play around with ideas and continued the blog. To the side are my posts, plus links to the old ones included in my book. I have also posted here the final draft of the book as I submitted it to the AMS HERE, including my submission for the cover that date from my and my student's work in the late 1990's. Needless to say, the book is also available at the pirate Russian website genesis.
Volume two of my collected works is now in the public domain. This contains about half of my papers in algebraic geometry as well many letters I received from Grothendieck. Volume one contains the remainder of my papers in algebraic geometry and remains under Springer copyright. It can be downloaded from National Taiwan University here. LINK
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AIs and Humans with Agency
I believe we are heading towards a major transformation of AIs. To date, the large language models (LLMs) have answered questions, written essays and solved well described problems. What they have not done is to make decisions affecting the real world. I would argue that even the phrase "real world" is not understood by LLMs: they have no senses and they have been fed writings conjuring up vast numbers of alternate worlds. Their "life" is to spew out responses to questions. If LLMs are sent out to businesses to perform tasks formerly assigned to human employees, they will need to possess agency, the power to act in the real world. More specifically this requires them to work together with other humans, to collaborate and plan with them, to read their desires and emotions even though they have no emotions themselves. None of this has been explored by the enthusiastic billionaire purveyors of this technology.
When I first got involved with AI, specifically with computer vision, I was inspired by David Marr's book Vision. Here he pioneered the idea that AIs and human brains needed to solve the same problems, so there should be a common "Theory of the Computation" which can be instantiated either in silicon or brain tissue. I found this convincing then and still do. I wrote a blog in 2020 with the title The Astonishing Convergence of AI and the Human Brain (still online -- go to "Old Posts") describing the parallels between LLM and the brain. This post aims to look at agency in the same way.
I begin this post with a description of the lengthy process during which humans acquire the skills to collaborate and plan while simultaneously connecting and activating their frontal lobes. This has been studied from both a psychological and a neurological point of view. I then go on to look at what LLMs are doing now and what they promise to do in the near future. Here we introduce Edward Ashford Lee's idea that we are coevolving with AIs. but will describe several unsuccessful attempts to add agency to LLMs. Finally, I describe a radical new architecture due to Yann LeCun that may pave the way to more successful agentic AIs.
Black hats and white hats again — defining evil
After I wrote "Black Hats and While Hats" and posted it on my website, I understand that it led to a perception of anti-semitism for some people. This makes me very sad. The problem is that nearly all people take sides and find one side, either Hamas's attack or Israeli bombing of Gaza, evil and unforgivable, the other side acting with justified self preservation. I tried to say neither side is wholly good nor wholly evil. I have always treated my teachers, colleagues, students and friends exactly the same, whether they were Jewish, Christian, Muslim, Hindu or Secular. I simply tried in that post to assemble the facts, as I knew or read them, and pose the huge moral dilemma over who has a claim to the small stretch of land "between the river and the sea", called Palestine.
With the exception of Buddhism, every religion seems to tell us mere humans that God wants us to do "good" actions, and avoid "bad" actions, especially "evil" ones. The foundation Judeo-Christian story is Eves temptation by Satan as snake to eat an apple from the Tree of Knowledge that God had forbidden. We bear the heavy burden of original sin. In all sects of the "religions of the book" (this includes Islam) there is no doubt that God has opinions about human actions and keeps track of what everyone does. So it is up to us humans to decipher God's wishes and use their God-given free will to act accordingly.
Letter to my Grandchildren-2
Dear Henry, Linus, Maya, Leela, Kaspar, Anarkali, Neerja and Arjuna.
Four years ago I wrote you what I thought were reasonable speculations and warnings about what I expect to happen in your future lives. I felt that major changes are going to happen in the 21st century and that I had a little insight, looking at the future from a scientific viewpoint. And I wanted to share this with you as it might be useful. It's a blog post with URL https://www.dam.brown.edu/people/mumford/blog/2020/Letter.html. In particular, I described the power of both artificial intelligence and of genetic engineering to transform your lives. Of course, the first has come true in spades in just a few years. I expanded this letter in the last chapter of my book "Numbers and the World, Essays on Math and Beyond" giving more details and references to buttress my arguments. But a number of new things have shocked me since then and made me now a good deal more pessimistic about the trajectory of 21st century culture. It may be worthwhile for everyone to know that some terrible cataclysm could happen. In this case, I hope I'm wrong. I am also well aware that pessimism is a well known trap that snares people who live a long time (I'm now 87). But humor me and read this blog post describing my chain of thought.
Consciousness, Robots, and DNA
This is a paper to appear in the Proceedings of the International Congress for Basic Science, based on my talk in Beijing on July 17, 2023. The talk explores the complex interplay between consciousness, robotics, and DNA using the insights of artificial intelligence, neuroscience and physics. Specifically, I discuss the evolution of language models and the quest for human-level AI, the scientific dimensions of consciousness, the elusive nature of "now" in consciousness, and the enigma of so-called cat-states in quantum theory. Finally, we examine the role of DNA replication as a possible generator of cat-states and its implications for understanding consciousness, free will, and the nature of reality. I hope to encourage further exploration at the intersection of these fields, highlighting the profound questions they raise about the nature of consciousness in both biological and artificial systems.
Black hats and white hats
In classic Hollywood westerns, you could always tell the villains from the good guys by the color of their hats. There was no doubt who was the struggling family trying to forge a living in the wild west and who was the villain trying to cheat him and her of the fruits of their labor, by murdering and pillaging. It is only human to seek clear cut moral judgements, to have unshakable convictions of what are allowable acts and what are hideous transgressions. But, in reality, this is simply not the way the world usually works. The history of Israel and of the Palestinians demonstrates the absurdity of summary judgements and the need for sympathy for both sides who are caught in an ever worsening spiral of hatred and killing. Unfortunately, almost everyone feels compelled to take sides. But from my perspective, there is a remarkable symmetry between the passions and fears of these two antagonists that compels sympathy for both. Both sides have feel that their continued existence in the "holy land" is threatened; their religions sanction war and they cannot see the humanity of the other side; they are locked in a spiral of hate. The terrorism of Hamas and the actions of Netanyahu's government have slammed the door shut on any 2 state solution. How did this ghastly situation happen?
Old Posts included in AMS Book
Here are links to pdfs of all my old posts that were. in modified form. included in my book Numbers and the World: Essays on Math and Beyond.
How to get middle school students to love a formula. 2014
Finding the Rhythms of the Primes. 2014
An Easy Case of Feynman's Path Integrals. 2014l
Can one explain schemes to biologists
Pythagoras's Rule
Is it Art?
Wake Up! (math publishing)
'All men are created equal'?
Math & Beauty & Brain Areas
Nationalism and the longing to belong. with best regards to Igor Shafarevich
Grammar isn't merely part of language
Hard to be an optimist (Population explosion)
Let the mystery be
Can an artificial intelligence machine be conscious?
Can an artificial intelligence machine be conscious. part II?
Letter to my Grandchildren<\a>
Ridiculous Math Problems
Reading Spinoza
The Astonishing Convergence of AI and the Human Brain
Nothing is Simple in the Real World
The Shape of Rogue Waves
Ruminations on cosmology and time
Like many people, I have been riveted for decades by the breathless bulletins from cosmologists describing the latest twist to their model of space and time at the largest possible scale. But recently, I have been looking more closely at these theories and, frankly, do not find them 100% convincing. Maybe it's all true but maybe in 50 years, it will all change. My biggest source of skepticism is its treatment of time: it feels as if in several ways it is trying to undo the vista that special and general relativity opened up for potential models of space-time, that the "standard model" reverts to a very Newtonian perspective on which an extremely simple relativistic model has been foisted. Let me explain.
The Dismal Science and the future of work
Economics is an area that is built on mathematical models that simplify highly complex phenomena. People often forget that, as a result, economic models omit human and historical factors that are fundamentally non-mathematical and outside its scope. Thus the impossibility of building mathematical models of human psychology undermines that basic building block of economics, the "rational economic agent". But I also want to argue that advances in technology are transforming society in ways not dealt with in economic models, by altering the need for most human work, another foundation stone of economics. My thesis in this post is that, in addition to dealing with the Malthusian constraints caused by population growth, the next 50 years will see the growth of a nearly completely automated society that requires only minimal work from the large majority of its citizens. Such a development destroys the basic axioms on which economics is built, not to mention the basic structure of human lives. How in heaven's name will we adjust to such a "gift"?
Recent Blogposts
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AIs and Humans with Agency
April 21, 2026 -
Black hats and white hats again — defining evil
January 20, 2025 -
Letter to my Grandchildren-2
August 30, 2024 -
Consciousness, Robots, and DNA
January 3, 2024 -
Black hats and white hats
December 12, 2023 -
Old Posts included in AMS Book
September 1, 2023 -
Ruminations on cosmology and time
March 1, 2021 -
The Dismal Science and the future of work
July 20, 2015
