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At MSFP 2008 in Iceland I chatted with Dan Piponi about physics and intuitionistic mathematics, and he encouraged me to write down some of the ideas. I have little, if anything, original to say, so this seems like an excellent opportunity for a blog post. So let me explain why I think intuitionistic mathematics is good for physics.
Intuitionistic mathematics, whose main proponent was L.E.J. Brouwer, is largely misunderstood by mathematicians. Consequently, physicists have strange ideas about it, too. For example, David Deutsch somehow managed to write in his otherwise excellent popular science book "The Fabric of Reality" that intuitionists deny existence of infinitely many natural numbers (those would be the ultrafinitists, if there are any). He also produced rather silly arguments against intuitionistic mathematics, which I explained to myself by believing that he never had a chance to learn that intuitionistic mathematics supports his point of view.
While Brouwer's and other preintuitionists' reasons for intuitionistic mathematics were philosophical in nature, there is today a vibrant community of mathematicians, logicians, computer scientists, and even the odd physicist, who work with intuitionistic mathematics not because of their philosophical conviction but because it is simply the right kind of math for what they are doing.
Intuitionistic understanding of truth
A common obstacle in understanding intuitionistic logic is the opinion that the difference between classical and intuitionistic logic arises because classicists and intuitionists just happen to disagree about what is true. A typical example of this is the principle known as Proof by Contradiction:
For every proposition $\phi$, if $\phi$ is not false then $\phi$ is true.
With a formula we write this as
$\forall \phi \in \mathsf{Prop}, \lnot \lnot \phi \Rightarrow \phi$.
Classical mathematicians accept it as true. Intuitionists do not accept it, but neither do they claim it is false. In fact, they claim that the principle has no counterexamples, that is
$\lnot \exists \phi \in \mathsf{Prop},
\lnot (\lnot \lnot \phi \Rightarrow \phi)$.
This becomes very confusing for classical mathematicians who think that the two displayed formulae are equivalent, because they believe in Proof by Contradiction. It is like believing that the Earth is flat while trying to make sense of Kepler's Laws of planetary motion.
The difference between intuitionistic and classical logic is in the criteria for truth, i.e., what evidence must be provided before a statement is accepted as true. Speaking vaguely, intuitionistic logic demands positive evidence, while classical logic is happy with lack of negative evidence. The intuitionist view is closer to the criterion of truth in science, where we normally confirm a statement with an experiment (positive evidence), but this analogy should not be taken too far.
What counts as "evidence" is open to interpretation. Before I describe the three most common ones below, let me just explain the difference between $\phi$ ("$\phi$ is true") and $\lnot \lnot \phi$ ("$\phi$ is not false"). Intuitionistically:
- $\phi$ holds if there is positive evidence supporting it,
- $\lnot \phi$ holds if it is contradictory to assume $\phi$, that is to say, evidence of $\phi$ would entail a contradiction.
- $\lnot \lnot \phi$ holds if it is contradictory to assume that it is contradictory to assume $\phi$.
That is a bit complicated. In essence, it says that $\lnot \lnot \phi$ is accepted when there is no evidence against it. In other words, $\lnot \lnot \phi$ means something like "$\phi$ cannot be falsified" or "$\phi$ is potentially true". For example, if someone says
"There is a particle which does not interact with anything in the universe."
that would be a statement which is not accepted as true, for how would you ever present positive evidence? But it is accepted as potentially true, for how would you ever falsify it?
A statement which is logically equivalent to one of the form $\lnot \lnot \phi$ is called doubly negated. For the purposes of this post I shall call a statement $\phi$ potentially true if its double negation $\lnot \lnot \phi$ is true. It seems nontrivial to come up with useful statement in physics which are only potentially true (but see the discussion about infinitesimals below). Perhaps Karl Popper would have something to say about that.
Let me now describe three most common interpretations of "evidence" in intuitionistic logic.
Computational interpretation
This is the interpretation of intuitionistic logic commonly presented in computer science. We view all sets as represented by suitable data structures—a reasonable point of view for a computer scientist. Then a statement is taken to be true if there exists a program (computational evidence) witnessing its truth. To demonstrate the idea, consider the statement
$\forall x \in A, \exists y \in B, \phi(x, y)$.
This is taken to be true if there exists a program which accepts $x$ and outputs $y$ together with computational evidence that $\phi(x,y)$ holds. Another example: the statement
$\forall x \in A, \phi(x) \lor \psi(x)$
is true if there exists a program which takes $x$ as input and outputs either $0$ and evidence of $\phi(x)$, or $1$ and evidence of $\psi(x)$. In other words, the program is a decision procedure which tells us which of the two disjuncts holds, and why. Under this interpretation the Law of Excluded Middle fails because there are unsolvable decision problems, such as the Halting problem.
The computationally minded readers might entertain themselves by figuring out a computational explanation of potentially true statements (Hint: first interpret Pierce's Law in terms of continuations). I have not done it myself.
Topological interpretation
We may replace the phrases "data structure" and "program" in the computational interpretation by "topological space" and "continuous function", respectively. Thus a statement is true if it is witnessed by a continuous function which transforms input (hypotheses) to output (conclusions).
The basis for this explanation may be found in physics if we think about what it means for a function to be continuous in terms of communication or information processing. Suppose an observer wants to communicate a real-valued quantity $x$ to another observer. They can do it in many ways: by making sounds, by sending electromagnetic signals, by sending particles from one place to another, by manufacturing and sending a stick of length $x$ by mail, etc. However, as long as they use up only a finite amount of resources (time, space, energy) they will be able to communicate only a finite amount of information about $x$. Similarly, in any physical process (computer, brain, abacus) which transforms an input value $x$ to an output value $f(x)$ the rate of information flow is finite. Consequently, in finite time the process will obtain only a finite amount of information about $x$, on the basis of which it will output a finite amount of information about $f(x)$. This is just the definition of continuity of $f$ phrased in terms of information flow rather than $\epsilon$ and $\delta$. Notice that we are not assuming that $f$ is computable because we do not want to make the rather sweeping assumption that all physical processes are computable.
The conclusion is that "all functions are continuous", including those that witness truth of statements.
You might be thinking that an analog-to-digital converter is a counterexample to the above argument. It is a device which takes as input an electric signal and outputs either 0 or 1, depending on whether the voltage of the signal is below or above a given threshold. Indeed, this would be a discontinuous function, if only such converters worked exactly. But they do not, they always have a tolerance level, and the manufacturer makes no guarantees about it working correctly very close to the threshold value.
A useful exercise is to think about the difference between "all functions are continuous", "potentially all functions are continuous", and "all functions are potentially continuous". Which one does the above argument about finite rate of information processing support?
Local truth
This explanation of intuitionistic logic is a bit more subtle, but also much more powerful and versatile. It is known by categorical logicians as the Kripke-Joyal or sheaf semantics, while most logicians are familiar at least with the older Kripke semantics.
Imagine a planet and a meteorologist at each point of the surface, measuring the local temperature $T$. We assume that $T$ varies continuously with position. A statement such as $T > 273$ is true at some points of the planet and false at others. We say that it is locally true at $x$ if there exists a small neighborhood around $x$ where it is true. In other words, a statement is locally, or stably true at a given point if it remains true when we perturb the point a little.
On this planet a statement is globally true if it is locally true everywhere, and it is globally false if its negation is locally true everywhere. There are also many intermediate levels of truth. The truth value (a measure of truth) of a statement is the set of those points at which the statement is locally true. Such a set is always open.
The explanation so far is a bit wrong. For a statement to be locally true at $x$, not only must it be true in a neighborhood of $x$, but it must also be true everywhere in the neighborhood "for the same reason". For example, the statement
$T > 273$ or $T \leq 273$
is true at $x$ if there exists a neighborhood $U$ of $x$ such that $T > 273$ everywhere on $U$, or $T \leq 273$ everywhere on $U$. The reason, namely which of the two possibilities holds, must be the same everywhere on $U$.
The truth value of $T = 273$ is the interior of the set of those points at which $T$ equals 273, while the truth value of $T \neq 273$ is the exterior of the set of those points at which $T$ equals 273. Thus the truth value of the disjunction
$T = 273$ or $T \neq 273$
need not be the entire planet—it will miss isolated points at which $T$ is 273. The Law of Excluded Middle is not valid.
By changing the underlying space and topology, we can express various notions of truth. We can, for example, incorporate passage of time, or a universe branching into possible worlds. In the most general case the underlying space need not even be a space, but a category with a so-called Grothendieck topology which determines what "locally" means.
Apart from being a wonderful mathematical tool, it should be possible to use sheaf semantics to clarify concepts in physics. I would expect the notions of "truth stable under small perturbation" and "truth local to an observer" to appeal to physicists. Fancy kinds of sheaf semantics have been proposed to explain features of quantum mechanics, see for example this paper by Bas Spitters and his coworkers.
Smooth infinitesimal analysis
Philosophical explanations and entertaining stories about intuitionistic mathematics are one thing, but getting actual benefits out of it are another. For physicists this means that they will want to calculate things with it. The good news is that they are already doing it, they just don't know it!
There is something odd about how physicists are taught mathematics—at least in my department. Physics majors learn the differential and integral calculus in the style of Cauchy and Weierstrass, with $\epsilon$–$delta$ definitions of continuity and differentiability. They are told by math professors that it is a sin to differentiate a non-differentiable function. They might even be told that the original differential and integral calculus, as invented by Leibniz and Newton, was flawed because it used the unclear concept of infinitesimals, which were supposed to be infinitely small yet positive quantities.
Then these same students go to a physics class in which a physics professor never performs $\epsilon$–$\delta$ calculations, freely differentiates everything in sight, and tops it off by using the outlawed infinitesimals to calculate lots of cool things. What are the students supposed to think? Clearly, the "correct" mathematics is useless to them. It's a waste of time. Why aren't they taught mathematics that gives a foundation to what the physics professors are actually doing? Is there such math?
Yes there is. It's the mathematics of infinitesimal calculus, brought forward to the 20th century by Anders Kock and Bill Lawvere under the name Synthetic Differential Geometry (SDG), or Smooth Infinitesimal Analysis. (I am too young to know exactly who invented what, but I've heard people say that Eduardo Dubuc also played a part. I would be happy to correct bibliographical omissions on my part.) By the way, I am not talking about Robinson's non-standard analysis, which uses classical logic.
This is not the place to properly introduce synthetic differential geometry. I will limit myself to a few basic ideas and results. For a first reading I highly recommend John Bell's booklet A Primer of Infinitesimal Analysis. If you refuse to read physical books, you may try his shorter An Invitation to Smooth Infinitesimal Analysis online. For further reading Anders Kock's Synthetic differential geometry is an obvious choice (available online!), and there is also Moerdijk and Reyes's Models of smooth infinitesimals analysis, which shows in detail how to construct models of SDG using sheaves of germs of smooth functions.
To get a feeling for what is going on, and why intuitionistic logic is needed, let us review the usual proof that infinitesimals do not exist. This requires a bit of logical nitpicking, so bare with me. Both intuitionistic and classical mathematics agree that there is no real number $x$ which is neither negative, nor zero, nor positive:
$\lnot \exists x \in \mathbb{R}, \lnot (x < 0) \land \lnot (x = 0) \land \lnot (x > 0)$.
(There is some disagreement as to whether every number is either negative, zero, or positive, but that is beside the point right now.) A nilpotent infinitesimal of second degree, or just infinitesimal for short, is a real number $dx$ whose square is zero. Any such $dx$ is neither negative nor positive, because both $dx > 0$ and $dx < 0$ imply $dx^2 > 0$, which contradicts $dx^2 = 0$. If $dx$ were also non-zero, we would have a number which is neither negative, zero, nor positive. Thus we proved that an infinitesimal cannot be non-zero:
$dx^2 = 0 \Rightarrow \lnot \lnot (dx = 0)$.
A classical mathematician will now conclude that $dx = 0$ by applying Proof by Contradiction. Intuitionistically we have only shown that infinitesimals are potentially equal to zero.
But are there any infinitesimals which are actually different from zero? It can be shown from the main axiom of SDG (see below) that non-zero infinitesimals potentially exist. It is a confusing world: on one hand all infinitesimals are potentially zero, but on the other non-zero ones potentially exist. Like all good things in life, intuitionistic mathematics is an acquired taste (and addictive).
Can a physicist make sense of all this? We may think of infinitesimals as quantities so small that they cannot be experimentally distinguished from zero (they are potentially zero), but neither can they be shown to all equal zero (potentially there are some non-zero ones). By the way, we are not talking about lengths below Planck length, as there are clearly reals numbers smaller than $1.6 * 10^(-35)$ whose square is positive.
The actual axiom which gets the infinitesimal calculus going does not explicitly state anything about non-zero infinitesimals. Instead, it expresses the principle of micro-affinity (sometimes called micro-linearity) that physicists use in their calculations.
Principle of micro-affinity: An infinitesimal change in the independent variable $x$ causes an affine (linear) change in the dependent variable $y = f(x)$.
More precisely, if $f : R \to R$ is any function, $x \in R$ and $dx$ is an infinitesimal, then there exists a unique number $f'(x)$, called the derivative of $f$ at $x$, such that $f(x + dx) = f(x) + f'(x) dx$. This principle has many consequences, such as potential existence of non-zero infinitesimals described above. For actual calculations the most important consequence is
Law of cancellation: If $a$ and $b$ are real numbers such that $a \cdot dx$ = $b \cdot dx$ for all infinitesimals $dx$ then $a = b$.
What this says is that we may cancel infinitesimals when they are arbitrary. This is important because infinitesimals do not have inverses (they are potentially zero). Nevertheless, we may cancel them in an equation, as long as they are arbitrary.
Let me show how this works in practice by calculating the derivative of $f(x) = x^2$. For arbitrary infinitesimal $dx$ we have
$f'(x) \cdot dx = f(x + dx) – f(x) = (x + dx)^2 – x^2 = x^2 + 2 x \cdot dx + dx^2 – x^2 = 2 x \cdot dx$
where we used the fact that $dx^2 = 0$. Because $dx$ is arbitrary, we may cancel it on both sides and get $f'(x) = 2 x$. I emphasize that this is a mathematically precise and logically correct calculation. It is in fact very close to the usual treatment which goes like this:
$f'(x) = (f(x+dx) – f(x))/dx = (x^2 + 2 x \cdot dx – dx^2 – x^2)/dx = 2 x + dx = 2 x$
There are two incorrect steps here: we divided by an infinitesimal $dx$ without knowing that it is different from zero (it isn't!), and we pretended that $2 x + dx$ is equal to $2 x$ because "$dx$ is very small". By the same reasoning we should have concluded that $f(x+dx) – f(x) = f(x) – f(x) = 0$, but we did not. Why?
The principle of micro-affinity allows us to easily derive the usual rules for computing derivatives, the potential existence of non-zero infinitesimals, prove the fundamental theorem of calculus in two lines, derive the wave equation like physicists do it, etc. And it is all correct, exact math. No approximations, no guilty feeling about throwing away "negligible terms" here but not there, and other hocus-pocus that physicists have to resort to because nobody told them about this stuff.
Just for fun, let me compute more derivatives. The general strategy in computing $f'(x)$ is to consider an arbitrary infinitesimal $dx$ and express $f'(x) \cdot dx = f(x + dx) – f(x)$ as a quantity multiplied by $dx$. Then we cancel $dx$ on both sides and get $f'(x)$. Throughout we use the fact that $dx^2 = 0$. Here we go:
- The derivative of $x^n$ is $n \cdot x^(n-1)$:
$(x+dx)^n – x^n = x^n + n x^(n-1) \cdot dx – x^n = n x^(n-1) \cdot dx$
- Leibniz's formula for derivatives of products $(f(x)\cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x)$:
$f(x+dx) \cdot g(x+dx) – f(x) \cdot g(x) = $
$(f(x) + f'(x) \cdot dx) (g(x) + g'(x) \cdot dx) – f(x) \cdot g(x) =$
$(f'(x) g(x) + f(x) \cdot g'(x)) \cdot dx$.
- Chain rule $f(g(x))' = f'(g(x)) \cdot g'(x)$
$f(g(x+dx)) – f(g(x)) =$
$f(g(x) + g'(x) \cdot dx) – f(g(x)) =$
$f(g(x)) + f'(g(x)) \cdot g'(x) \cdot dx – f(g(x)) =$
$f'(g(x)) \cdot g'(x) \cdot dx$
where we used the fact that $g'(x) \cdot dx$ is infinitesimal because its square is zero.
There you have it, in a paragraph we derived precisely and in sufficient detail what usually takes a whole lecture of $\epsilon$–$\delta$ manipulations.
If we stick to classical logic, the Principle of micro-affinity is false. To see this, consider a function with a jump, such as
$j(x) = 0$ if $x < 0$
$j(x) = 1$ if $x \geq 0$
At $x = 0$ the principle of micro-affinity fails. This is a counterexample only in classical mathematics because intuitionistically we cannot prove that there is a function with a jump. Concretely, the above definition of $j(x)$ is not intuitionistically valid because it presupposes $\forall x \in R, x < 0 \lor x \geq 0$.
Space-time anomalies
But wait! Intuitionistically we can construct non-differentiable continuous functions, such as the absolute value $f(x) = |x|$, for which the principle of micro-affinity fails, too. Well, I am not telling the whole story. The smooth real line $R$ of infinitesimal analysis is not the usual real line $\mathbb{R}$, as constructed by Richard Dedekind. It does not support computation of absolute values.
This seems pretty bad. If we cannot have a function such as the absolute value, then it is not clear how to model phenomena that involve sudden change of direction, such as reflection of light and collision of particles. Can rays not bounce of mirrors, intuitionistically? Yes they can, it is just that the intuitionistic treatment of sudden changes is more profound than the classical one.
Consider a particle which moves freely up to time $t_0$, then bounces off a wall, and moves freely after that. Its position $p$ is described as a function of time $t$ in two parts,
$p(t) = p_1(t)$ if $t \leq t_0$
$p(t) = p_2(t)$ if $t \geq t_0$
where $p_1$ and $p_2$ are smooth functions, and $p_1(t_0) = p_2(t_0)$. Because $p$ is defined separately for $t \leq t_0$ and for $t \geq t_0$, its domain of definition is the union of two half-lines
$D = \lbrace t \in R \mid t \leq t_0 \rbrace \cup
\lbrace t \in R \mid t \geq t_0 \rbrace$.
Classical mathematics proves that $D = R$, which amounts to forgetting that $t_0$ is a special moment. In the smooth world, $D$ is only a subset of $R$, but is not equal to $R$ because it carries more information than $R$. As strange as this may seem, it is useful because it encodes moments in time or places in space where special things happen, such as sudden change of movement or sudden change of density. Smooth space-time, say $R^4$, allows only smooth motion and smooth distribution of mass. If we place non-smooth mass in it, the space will change to a subset of $R^4$ which carries additional information about the anomalies contained in it.
This post has become very long so I will stop here.

subset: commoncrawl


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Svein Tuft Time Trials to Victory in San LuisFri 25 Jan 2013
Svein Tuft time trialed to the first international victory for ORICA-GreenEDGE during the Australian outfit's second season. The eight time Canadian national time trial champion set a scorching speed of 22:14 over 19.2 kilometres. Tuft edged out Leandro Messineo (San Luis Somos Todos) by seven seconds to win stage four of Tour de San Luis. Michal Kwiatkowski (Omega Pharma-QuickStep) rounded out the stage podium.
"This time of year, I'm not always sure what I'm going to get," admitted Tuft. "I've had good feelings, but I was a little surprised at how hard it felt when I was out there. I forget every year how hard it is. I think that's what allows me to keep doing it."
Tuft credits ORICA-GreenEDGE trainer Marc Quod with giving him an extra edge.
"I did a lot of strength and endurance in the off season," Tuft said. "Working with Quod has been really good. It's probably been one of the better winters for me. I've focused on things that I don't normally work. We kept the training really simple and I think that helped."
"Another fortunate thing is that I came down early to Argentina," he added. "I was able to adjust to the heat and riding in the local climate. That's definitely been to my benefit."
The non-technical course suited Tuft's strengths.
"It was a wide open highway-type course," he said. "It included two roundabouts and a couple corners. That's it. It was really simple in that sense. The wind was a bit difficult going out, and the downhill sections were super fast. It's definitely a course made for me."
Tuft carefully metered his efforts during the first half of the time trial and flew home during the second half.
"The sections with the gradient were tough," he explained. "I had to really put myself out there without going too far into the red too early. It was definitely important not to lose any time on the way out. I pushed hard and recovered where I could with the momentum I had already built. I think that really paid off for me. There are strong guys here, and I had to give it everything I had."
Tuft ticked off a few key time trials he plans to target during the remainder of the season – specifically mentioning both the team time trial and individual time trial at Tirreno Adriatico and the individual time trial at Circuit de la Sarthe. Halfway through his list, he stopped himself with the realization that he targets every time trial he starts.
"I look to take advantage of every opportunity to time trial," he said. "There's never any guarantee of good form, so when I have it, I'm ready to make the most of it."
Looking at Tour de San Luis as a whole, Tuft has been satisfied with his first experience racing in Argentina.
"This is a really great race to start out the year," said Tuft. "It's not super high stress, but it is super high quality. Everyone is here to race their bikes. For January, it's a really good race. I'm glad the team decided to race here this year."


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President Obama, whom college kids overwhelmingly favored in Tuesday's election, vowed on the campaign trail to increase federal financial aid. But experts say a divided Congress could put those changes on hold — and lead to higher college costs for many families.
Automatic spending cuts that are set to kick in at year-end — as part of the so-called fiscal cliff — could result in an 8% cut in federal aid. Obama's aides have said the president plans to move quickly to find common ground with Republicans over the $1.2 trillion in overall cuts.
If they do begin as scheduled, however, cuts would come at a time when federal financial aid is already on the decline. According to a report released last month by the College Board, federal grants totaled $49 billion during the last academic year, down 5% from a year prior, after adjusting for inflation — the first drop after rising for five consecutive years. Federal work study, which allows colleges to create campus-based jobs for students, fell 4% to $972 million, after adjusting for inflation, the first time it's dropped below the $1 billion mark in at least a decade.
And college experts caution that even more cuts could kick in starting next fall. Next year, the Higher Education Act, a major piece of legislation that for decades has helped determine how much college aid — including grants and loans — the federal government provides, is up for reauthorization. If it isn't renewed next year, there's an automatic one-year extension of the current rules giving students one more academic year of breathing room before changes kick in.
• Barack Obama wins another four years
• Obama victory sets up the next battle
• Five things the Republicans must do now
• Time for Obama to show leadership on taxes
• Three global tasks for the second term
• Democratic wins may thwart grand bargain
• Three post-election concerns for investors
• Obama: The best is yet to come for America
• Romney urges an end to political bickering
But some experts say even the best-case scenario will leave families worse off than this year. That's largely because no matter what happens to financial aid — even if it stays the same or increases — it won't keep up with the rate at which tuition is rising, says Mark Kantrowitz, publisher of FinAid.org, which tracks financial aid issues.
While there's still time before any of these decisions are made, college advisers say families should prepare for the possibility of less aid by broadening their financial aid search from now. That includes contacting colleges' financial aid offices for school grants and searching for scholarships in their community and on sites like Fastweb.com (Kantrowitz is also publisher of this site) and CollegeBoard.org. High-school seniors might also want to consider applying to public in-state colleges, since their tuition costs are significantly lower than private colleges.
In the meantime, here are three types of financial aid that could see changes in 2013.
Pell grants
About 9 million undergraduates received federal Pell grants to help cover tuition costs in 2010-11, according to the latest data from FinAid.org. The latest Consolidated Appropriations Act that was signed into law last December, which set most federal agencies' budgets for 2012 (and was later extended to March 2013), scaled back spending for this grant program.


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Investigator Program
HHMI catalyzes discovery science by investing in outstanding researchers with generous and flexible funding for significant periods of time. HHMI selects people, not projects. Our Investigators ask tough questions in science, even at the risk of failure. They develop new tools and methods that make it possible to ask previously unapproachable questions. Many forge links between biology and medicine. They are committed to mentoring and training the next generation of scientists.
With HHMI's backing, scientists have radically changed our understanding of cells, the brain, the immune system, the development of organs, and the causes of human diseases. To date, 29 current or former HHMI Investigators have been awarded the Nobel Prize.
HHMI employs and supports nearly 300 Investigators, who are based at over 60 universities, colleges, and other research institutions across the U.S. In addition to their research programs, HHMI Investigators participate in educational activities and hold leadership roles at these host institutions. With our backing, as well as support from other sources, Investigators direct a research team and create a laboratory environment well suited to tackle profound science questions.
HHMI Investigator Attributes
HHMI Investigators have a combination of the attributes listed below. These attributes are used both in review of applicants during competitions and during review of Investigators for appointment renewals:
- They identify and pursue significant biological questions in a rigorous and deep manner.
- They drive their chosen research field into new areas of inquiry, being consistently at its forefront.
- They develop new tools and methods that enable creative experimental approaches to biological questions, bringing to bear concepts or techniques from other disciplines when necessary.
- They forge links between basic biology and medicine.
- They demonstrate great promise of future original and innovative contributions.
- They are active in service and training in their host institutions, and in the greater scientific community.
Eligibility Criteria
Through periodic competitions, HHMI invites applications from scientists who meet the eligibility criteria. Although specific criteria may vary for each competition, in general:
- Applicants must hold a PhD and/or MD.
- Applicants must plan to perform research at one of over 200 eligible U.S. institutions.
- Applicants must hold a position that represents substantial commitment by their institution, such as a tenure-track faculty position.
- Applicants must have a track record of peer-reviewed funding.
- Each competition defines a required length of post-training professional experience prior to the application.
- Each competition defines the breadth of scientific disciplines to be considered; however, most competitions are open to all biomedical disciplines.
Application and Competition Information
Through periodic competitions, HHMI invites applications from eligible scientists. Subscribe to HHMI News to receive updates on future competitions.
- Applicants apply directly; there is no nomination process and no limit of the number of applications from a single institution.
- Application requirements include a CV, five highlighted research articles, and a program description focused on research achievements during the last five years and future research plans.
- Selection of Investigators is made by HHMI scientific leadership and is based on assessment by a panel of distinguished scientists, including members of HHMI advisory boards.
The HHMI Investigator appointment has a seven-year renewable term. Renewal decisions are made by HHMI scientific leadership, based on assessment by a panel of distinguished scientists, including members of HHMI advisory boards. Read more about the review process »
- Investigators are employees of HHMI and receive salary and benefits from the Institute.
- Investigators are provided generous financial support for research as well as access to capital equipment funds.
- Investigators belong to a community of nearly 300 Investigators as well as other HHMI-funded scientists. They annually attend one of several scientific meetings that highlight science within the Institute.
Appointments involve a collaborative agreement with the Investigator's host institution, where the researcher holds a concurrent appointment and the research is performed.
HHMI Investigators
The HHMI Investigator program is the Institute's flagship program aimed at sustaining innovation in science.


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pretty cool but there is better
I think this is quite interesting but I think it would be better if the system tracked eyeball movement and then moved the camera in response.. obviously far more difficult.
A more interesting project I saw before on hackaday was where a guy mounted a camera off the back of his shoulders and set up slightly and wore a pair of goggles. He was literally watching himself walk around as if he was in something like second life.
Infact.. video here: http://www.marcowens.co.uk/avat.html