David Mumford

Archive for Reprints, Notes, Talks, and Blog

Professor Emeritus
Brown and Harvard Universities
David_Mumford@brown.edu

Blog

Below are my posts, from the most recent to the earliest. The blog is now set up via 'DISQUS" for uploading comments but I also welcome comments sent to me at dbmumford@gmail.com. I would like to post some these comments with my replies unless you ask that they be kept private. I won't post any comments sent under a pseudonym however: nothing on my blog justifies hiding. Subscriptions to the blog can be made through the RSS feed.
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Ruminations on cosmology and time

March 1, 2021

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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 Astonishing Convergence of AI and the Human Brain

October 1, 2020

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One of the earliest ideas for programming Artificial Intelligence was to imitate neurons and their connectivity with an architecture called neural nets and an algorithm known as back propagation . In the turbulent boom and bust evolution of AI, this remained a theme with strong adherents, e.g. Geoffrey Hinton and Yan LeCun, but it fell out of the mainstream. But all this changed around 2010 when these ideas were implemented with really huge datasets and really fast computers. The field of AI has now had a decade of tremendous progress in which neural nets, along with some major improvements, have been the central character. The purpose of this post is to describe the parallels between the software implementation of AI and the human instantiation of intelligence in what is jokingly called "wet-ware", namely our brains. I find it startling how much this progress has been connected to better models of the brain. It's my impression that, for better or for worse, all future instances of artificial intelligence will be driven to use similar structures, possibly with some enhancements. One drawback is the degree to which these algorithms are opaque and resist simple explanations of why they do what they do. Alarmingly, this suggests that we will also not be able to understand why future robots decide to do whatever they do.

Nothing is simple in the real world

July 8, 2020

Tags: future, and more

Mathematicians and politicians have one thing in common: they both thrive on oversimplifying aspects of the real world. If there is one thing that a college education should teach every student it is that there are two sides to every question, that the world is not simple, that good and bad are not reducible to checking who is wearing a white hat, who a black. Mathematical models are based on isolating some effects, some aspects of a very complex real world situation and modeling this simplification. Political rhetoric is pretty much the same. The sad thing is that not only left and right wing activists vastly oversimplify the challenges our society faces but that now radical students at elite universities are so carried away by partial insight that they lose sight of the ideals of an open society. In this short post, I want to wade into two heavy duty issues that are often over-simplified: slavery vs. human rights and eugenics, the discussion of which provokes the strongest emotions. They are related as both have been supported by racism, justifying the enslavement of another race and regarding any people of mixed blood as equally inferior.

April 19, 2020

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In our secular age, it is hard to bridge the gap between the long tradition of theistic philosophers and contemporary science-based speculation about the nature and fate of humankind. My friends are all over the map -- from avowed atheists to weekly church-goers. I have not been a regular churchgoer since graduating from Phillips Exeter where Sunday church attendance was compulsory. The word "God" was already an obstacle for me as the idea of him as a super-powerful old man in white robes watching and judging every action of every human felt so absurd, there seemed no point in looking further. But in the back of my mind, I knew that all those famous thinkers in the Judeo-Christian tradition were far from stupid. Struggling to find my own path, I stumbled last year upon Spinoza and, to my surprise, found a great deal that I could understand, though not without a struggle. This post is about my efforts to understand his writings and to understand their relation to other key ideas in my thinking, e.g. Plato, Descartes, Buddhism and Physics.

Ridiculous Math Problems

April 1, 2020

Tags: future, and more

Maybe some humor is useful while we are all caught up in the depressing whirlwind of this Covid-19 pandemic. Recently, a silly math problem went viral on facebook about what appear to be prices for a doll, a pair of shoes and a pair of scarves (reproduced below). It is an example of how the public loves brain-teasers, odd puzzles with some math in them. What makes this interesting to me is that this playing with really meaningless math problems is something that mathematicians do too. There is an ancient tradition of ridiculous math problems that permeate the history of math, especially the history of algebra. I find it odd that no book on the History of Math points out how many algebra problems in every era are crazy concoctions whose main point is to show how smart their creator was or how nifty their discoverer's new tool is. It's a fascinating, not well-known side of math.

Letter to my Grandchildren

March 1, 2020

Tags: future, and more

Dear Henry, Linus, Maya, Leela, Kaspar, Anarkali and Neerja, and I'm not forgetting my step-grandchildren, Hannah, Gordon, Courtney, Jake, Kate, Emily, and Nina

I worry a great deal about how you, my offspring after two generations, are going to fare in your lifetimes, living in this remarkable world in which we find ourselves. Putting aside for a bit guesses about the various things that may or may not happen, one thing seems certain: both the physical world we are living in and the culture by which we live in it are changing very fast, arguably faster than they ever did during the entire history of mankind.

Can an artificial intelligence machine be conscious, part II?

July 12, 2019

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Since posting "Can an artificial intelligence machine be conscious", I have read two books which argue for the belief that conscious artificial intelligences are on their way and we should prepare ourselves. I want to discuss both books and evaluate my own arguments in their light.

Can an artificial intelligence machine be conscious?

April 11, 2019

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The theory of artificial intelligence, during my lifetime, has gone through half a dozen cycles of boom and bust: periods when it was said confidently that computers will soon attain human level intelligence and periods of disillusionment when this seemed nearly impossible. Today, we are in the latest boom period and some visionary computer scientists are going even further, asking when "AI"s (using the abbreviation AI to make the machine sound like a new life-form) will actually attain not merely human intelligence but possess our consciousness as well.

Let the mystery be

April 13, 2018

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Iris DeMent sings, in her unique southern twang, a lovely song, (you can find it here on youtube.) Its chorus goes:

Everybody's wonderin' what and where they they all came from.
Everybody's worryin' 'bout where they're gonna go when the whole thing's done.
But no one knows for certain and so it's all the same to me.
I think I'll just let the mystery be.
I'm sure nothing I have to say about the deep questions of life is original. My recently deceased wife Jenifer was never predisposed to reflect like this. She truly lived in the present and withstood a progressive disease for 9 years by accepting each limitation as it came, sustained by friendships and love. But my curse, like many scientists, is to seek answers to questions that have no answers and writing my thoughts down seems to be part of the process. This post is a meditation on consciousness, the mind/body problem and what science has said about these issues, and then touches at the end on the nature of the soul.

Hard to be an optimist

January 1, 2017

Tags: population, pessimism, urbanization, pollution, and more

It's New Year's Day and the paper is full of predictions. The question of the day is how 2017 will go and especially can the U.S. survive the Trump earthquake? But, being a scientist, I find myself always trying to think longer term -- what are the basic problems the world faces, the roots of the issues that fill the papers? Will my grandchildren likely grow up in a good and prosperous world as I dearly wish? Unfortunately, I am filled with foreboding and I don't think this is just an old man's gloom. More precisely, I think essentially all the problems we face can be traced to one basic cause: the explosive increase of the human population -- Malthus's famous contention in An Essay on the Principle of Population. World population has increased by a factor of 3.4 in my lifetime. Recycle, buy solar panels -- fine, but nothing any of us can do is going to control our vast and still growing numbers and all the problems this unprecedented multitude brings.

Grammar isn't merely part of language

October 12, 2016

Tags: Grammar, Vision, Tom Wolfe, Chomsky, Everett, Phil Lieberman, and more

This post is inspired by reading the latest Tom Wolfe diatribe, "The Kingdom of Speech". While the book sets off to discuss the issues of what were the origins and evolution of speech in early man, the largest part of this book is devoted to a juicy recounting of the feud between Noam Chomsky and Daniel Everett over whether recursion and other grammatical structures must be present in all languages. Chomsky famously holds that some mutation endowed early man with a "language organ" that forces all languages to share some form of its built-in "universal grammar". Everett, on the other hand, was the first to thoroughly learn the vastly simplified language spoken by the Amazonian Piraha (pronounced peedahan) that possesses very little of Chomsky's grammar and, in particular, appears to lack any recursive constructions (aka embedded clauses). What I want to claim in this blog is that both are wrong and that grammar in language is merely a recent extension of much older grammars that are built into the brains of all intelligent animals to analyze sensory input, to structure their actions and even formulate their thoughts. All of these abilities, beyond the simplest level, are structured in hierarchical patterns built up from interchangeable units but obeying constraints, just as speech is.

Nationalism and the longing to belong, with best regards to Igor Shafarevich

September 15, 2016

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When I recently packed up my office files, how I could resist re-reading some of the old letters in my files? In 1992, there was a major controversy at the US National Academy of Science over censuring their Foreign Member Igor Shafarevich for anti-semitic writings and actions. Added Feb.20, 2017: I just learned that Igor passed away yesterday in Moscow. Let me now dedicate this post to his memory. He was an old friend and we exchanged quite frank letters at that time (his letter to me is appended at the end of this post). I did not believe then and do not believe now that he was anti-semitic, but rather that he was a fervent believer in his country, its people and its traditions -- perhaps one should say its soul. Now, 24 years later, in a world with right wing movements sweeping the entire world and with now the US President, I found many things in that controversy very relevant today and it made me rethink my own beliefs. Let me explain.

Math & Beauty & Brain Areas

October 11, 2015

Tags: beautiful-mathematics, brain localization, and more

Two questions about the general nature of math have a certain interest to the larger public: mathematicians like to talk of a 'beautiful result' but what does this mean?; and is there a special part of cortex which is highly active when people do math? Recently Professors Michael Atiyah and Semir Zeki have put these two questions together, collaborating on an astonishing experimental investigation of these questions entitled "The experience of mathematical beauty and its neural correlates". Fifteen mathematicians were scanned using fMRI while viewing 60 mathematical formulas and rating them as ugly, neutral or beautiful. All 60 equations are shown in the sidebar, in 6 jpegs: you can magnify them by clicking. Their main result is that activity in the mOFC = medial (near the centerline) orbital (in the inward curl of the cortex above the eyes) frontal cortex correlates to some extent with their judgement of beauty (though strangely activity in mOFC relative to baseline diminishes). My aim in this post is to argue for the view that the subjective nature and attendant excitement during mathematical activity, including a sense of its beauty, varies greatly from mathematician to mathematician and that that would make it plausible for quite different parts of the brain to be active during mathematical reflection. I do not claim any scientific basis for this as my only evidence comes from opportunities to talk with colleagues and being struck with the remarkably diverse ways they seem to have of 'doing math'.

The Dismal Science and the future of work

July 20, 2015

Tags: economics, work, technology, employment, and more

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"?

'All men are created equal'?

June 16, 2015

Tags: dalits, Jefferson, Manu, politics, India, caste, and more

In the Declaration of Independence, Jefferson famously wrote "We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness." Anyone with a brain knows how much these ringing words have been flagrantly ignored, certainly in the US and over pretty much the whole rest of the world. For some years I have been helping Prof. Shankar of the Chennai Mathematical Institute support an orphanage for Dalits (aka untouchables, "Harijans", scheduled castes) near Chennai in India and following to some extent his bulletins on the horrors endured by most Dalits and how they have none of the above rights. I stuck my neck out a few weeks ago writing a letter criticizing the "de-recognizing" of an activist pro-Dalit student group by the Director of the Indian Institute of Technology in Madras. The large reaction to this letter drew my attention to the extent to which this is a huge issue in India, so the purpose of this post is to look at the Indian struggle between the caste system and the ideals embodied in Jefferson's challenge.

Wake Up!

April 1, 2015

Tags: publishing, journals, Klaus Peters, Springer, private equity, and more

The world of professional publishing, of scholarly communication, is in a state of profound transformation. In some fields, for example physics and computer science, researchers have embraced this transformation and are forging new policies and better customs. In my experience, however, mathematicians are one of the most conservative research communities, clinging to old habits in spite of the opportunity to improve their working life. The impetus for this blog at this time is the death of Klaus Peters, a publisher who, more than any other that I have met, saw publishing in mathematics as a service to the professional community and strived tirelessly to find new ways to assist our community. But the changes that have happened in the commercial publishing world deeply disturbed him. I want to make a plea to my colleagues to spend more time considering how we should shape this aspect of our profession and then being open to radical changes: you have nothing to lose but the chains that are binding you to capitalist exploitation and you can gain a freer, simpler world to work in.

Is it Art?

February 9, 2015

Tags: art, Wolfe, Rockmore, beautiful-mathematical-expressions, and more

A couple of years ago, my good friend Dan Rockmore sent me by FedEx a remarkable invitation: write on copper plate "what you think is your most significant and elegant equation", for a limited edition of etchings. Even for Dan whom I've known for unorthodox projects, this seemed off the wall. But OK: Yole Zariski, the wife of my PhD advisor Oscar Zariski, had her artist brother cast some of Oscar's results on a necklace that she loved. Maybe the odd symbols that we put together might be viewed as a contemporary form of magic and, even if not understood, with a McLuhan-esque significance. And now, I learned from Dan, the project has expanded and they are seeking everyone's "beautiful mathematical expressions": login here "

Pythagoras's rule

January 9, 2015

Tags: Pythagoras, discovery, geometry, Mesopotamia, China, India, transmission, proof, and more

The earliest extant documents that show knowledge of the rule relating the lengths of the three sides of a right triangle, that is traditionally named after Pythagoras, are Babylonian tablets dating from the centuries around Hammurabi's time, c. 1800 BCE. I am calling it a rule, not a theorem, following Jens Høyrup's suggestion, because it appears as a rule for connecting these lengths, not a theorem, in most of its early history. In any case, we don't know if Pythagoras proved it or not. After the Babylonians it next appears in extant records in Indian Vedic altar construction manuals, composed and transmitted orally as early as 800 BCE. Due to the wholesale destruction of documents in China in the Qin dynasty (221-206 BCE), the earliest records we have for the rule from China date from the second century BCE. This is a sparse set of sources indeed. But because this rule may be described in math-talk as the first "non-trivial" mathematical theorem to be discovered, there has been extensive debate about when and where it was first found, whether it was discovered independently in several places and how it was found. All this work belongs to what André Weil called "protohistory", an attempt to be scholarly when surviving documents are not only sparse but also possibly unrepresentative of a tradition, and totally absent from other cultures. The full history of Pythagoras's rule is a perfect example of a problem on which one can only speculate. But that's what I want to do in this blog. However, all is not speculation and, for great help in all that a real scholar of the History of Math might study, I want to thank Jens Høyrup for all his help.

Can one explain schemes to biologists

December 14, 2014

Tags: Grothendieck, schemes, Nature-magazine, and more

John Tate and I were asked by Nature magazine to write an obituary for Alexander Grothendieck. Now he is a hero of mine, the person that I met most deserving of the adjective "genius". I got to know him when he visited Harvard and John, Shurik (as he was known) and I ran a seminar on "Existence theorems". His devotion to math, his disdain for formality and convention, his openness and what John and others call his naiveté struck a chord with me.

So John and I agreed and wrote the obituary below. Since the readership of Nature were more or less entirely made up of non-mathematicians, it seemed as though our challenge was to try to make some key parts of Grothendieck's work accessible to such an audience. Obviously the very definition of a scheme is central to nearly all his work, and we also wanted to say something genuine about categories and cohomology. Here's what we came up with:

An Easy Case of Feynman's Path Integrals

November 1, 2014

Tags: quantum-mechanics, quantum-computers, linear algebra, heat-bath, density matrix, and more

Like many pure mathematicians, I have been puzzled over the meaning of Feynman's path integrals and put them in the category of weird ideas I wished I understood. This year, reading Folland's excellent book Quantum Field Theory -- A Tourist Guide for Mathematicians, I got a glimmer of what was going on. In a seminar on quantum computers with my good friend John Myers a few years ago, I had played with finite dimensional quantum systems, so it was natural to work out Feynman path integrals in the finite dimensional case.What emerged was so clean and even undergraduate-linear-algebra ready, that I want to put this rigorous and simple result in my blog. A similar path was taken by Ben Rudiak-Gould in a recent arxiv submission "The sum-over-histories formulation of quantum mechanics".

The lowest zeros of Riemann's zeta are in front of your eyes

October 30, 2014

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Barry Mazur and William Stein are writing an excellent book entitled What is Riemann's Hypothesis? The book leads up to Riemann's 'explicit formula' which, in von Mangoldt's form, is the formula for the discrete distribution supported at the prime powers: $$\sum_{\text{primes }p} \sum_{n \ge 1} \log(p) \delta_{p^n} (x)= 1 - \sum_k x^{(\rho_k-1)} - \tfrac{1}{x(x^2-1)}$$ where $$x > 1$$, $$\rho_k$$ ranges over the zeros of the zeta function in the critical strip $$0 < \text{Im}(\rho) < 1$$ and the sum over k converges weakly as a distribution. This relates primes to the zeta zeros. In this blog, I ask: can we find the first and maybe more low zeroes hidden in the very smallest primes?