Sir Karl Popper says that products of our mind, like speech, music, math,... are things on their own. We create them. We put effort in them. I wonder where the difference of creating these things, is compared to let's say lifting a stone of mass m to height h. We also put measurable effort in there: E=mgh.

Can we measure the effort to create a mathematical theory in terms of energy?


From discussions in the comments, I feel the need for some clarifications. The term "mathematical theory" might be misleading. I like to measure the effort that is put into a "mathematical proof", like Wiles' proof of Fermat's Last Theorem.

What comes closest to what I want is the minimum amount of energy (we neglect dissipative processes) a computer would need to proof a theorem. There are some examples of Automated theorem proving.

Ok and now I'm speculating: Would it boil down to calculate the number of bits and transformations of bits that are necessary to perform every step of the proof and connect this to the amount of energy that is stored in a bit (see Information is physical)?

  • related but unsatisfying: Is Information a potential or kinetic kind of Energy?
    – draks ...
    Commented Jan 8, 2014 at 23:57
  • What's wrong with taking human metabolic energy as a proxy? (Or one step further: just call it labor.)
    – Rex Kerr
    Commented Jan 9, 2014 at 0:33
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    There is this story around about the amount of calories that get burned by a worldcup-chessmatch. It was absurdly high. Maybe the amount of burned calories is an indicator for the effort to create a mathematical theory.
    – Lukas
    Commented Jan 9, 2014 at 8:42
  • 1
    The only thing I a quick googlesearch turned up was this: news.stanford.edu/news/2009/june17/classday-061709.html "And if it’s the right wood and the right chess grand masters in the middle of a tournament, they are going through 6,000 to 7,000 calories a day thinking, turning on a massive physiological stress response simply with thought and doing the same thing with their bodies as if they were some baboon who has just ripped open the stomach of their worst rival, and it’s all with thought, and memories and emotions."
    – Lukas
    Commented Jan 9, 2014 at 9:28
  • 1
    Why isn't the "amount of burned calories" sufficient? After all if you take careful measurements inside a computer, the only observable physical output of a computation is heat.
    – user4894
    Commented Jan 29, 2014 at 0:21

4 Answers 4


I suspect that we could. Although such a measure may not be especially informative or useful. Consider for example that some brains may use more or less energy than other brains use when completing the same tasks. Nevertheless, it appears your question asks whether it is possible to measure the amount of effort expended in the creation of a single mathematical theory by measuring the amount of energy consumed in the creation of that theory.

The American Heritage English Dictionary defines work as

Physical or mental effort or activity directed toward the production or accomplishment of something.

In precise usage, the term work denotes the amount of force applied over a distance, and is commonly expressed Fd. But in any case, the product of that formula always represents an amount of energy. Accordingly, that product is normally expressed in joules.

The conversion of joules to calories is straightforward: 1 joule = 0.239 calories.

Thus, calories = joules = work = mental effort, and so by hypothetical syllogism, calories = mental effort.

As others have said, we can measure the number of calories a brain consumes over a period. So then one could measure the number of calories the mathematician's brain consumed while he was developing his theory (and, perhaps, to make the measure more informative, compare that to the amount of calories his brain consumes while at rest).

So the answer is, yes we can measure the amount of effort used to create a mathematical theory in terms of energy.

I'm going to abstract this and call the brain a box. To get a minimum amount of energy you would need to define the properties of the box and its starting state. More specifically,

  • What information does it begin with
  • How is this information arranged
  • How do the box's thinking processes proceed
  • What are the physical properties of the box (for example, its electrical resistance)

You need that information to answer your question. You could suppose some 'perfectly efficient mathematician', but in any case you would need to state your parameters.

The bottom line is that thinking uses energy. You remarked that the amount of energy used depends on the person creating the proof. So,

  1. If you want to measure the energy a particular person would use, then you can measure the work-rest difference as I had suggested.
  2. If you want to measure the absolute minimum energy needed for any possible brain to create a given proof then your answer is something infinitely close to 0 - just suppose perfectly conductive neurons, or a brain-state as it would be an instant before concluding the proof.
  3. If that is too absurd, then you would have to stipulate your starting conditions and your parameters so that they describe a more typical brain.

That said, it is also important to stipulate what information we have to answer the question? Do we have perfect knowledge? If so, then you know how much energy each person would use developing a given proof, control for the energy consumed at rest and thinking about the proof. The difference multiplied by the time spent thinking about the proof and only about the proof will be the energy used to create the proof. Consider the amount of energy each person consumed in formulating the proof, select the lowest value, and you have the minimum figure for all extant people.

  • 1
    Thanks for your effort. I'm more interested in the minimal value. Calories used by the brain just seem to be an upper bound...
    – draks ...
    Commented Jan 29, 2014 at 10:23
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    Assume your smarter than me (you should, you are Hal), then my difference might be higher than yours. I'm interested in the minimium...
    – draks ...
    Commented Jan 29, 2014 at 21:23
  • 1
    Sorry for the confusion, needed is correct. Your box idea is fine, but in fact I don't care about about any kind of brain. I like to connect an energy value to a mathematical proof, irrespective which brain is actual going through the proof and by that generates more or less overhead.
    – draks ...
    Commented Jan 30, 2014 at 12:28
  • 1
    Actually, I upvoted.
    – Hal
    Commented Jan 30, 2014 at 14:10
  • 1
    Hal, thanks for your thoughts. I think I gotta go back the lab and think about all this. Maybe energy is not the right measure for proof complexity...
    – draks ...
    Commented Jan 31, 2014 at 8:09

Of course not all parts of thinking necessary to solve a mathematical puzzle are conscious. You may solve the main trick in your dreams, and elaborate the complete answer when you awake. That's very common, actually. Anyway we can measure thinking, conscious and unconscious, at least in theory, because the brain is a kind of muscle, actually. If you study the evolution of invertebrate animals, you'll see that the same cells worked as skin, muscles and neurons (in Porifera and Cnidaria, for instance). When you do a EEG, you are registering electrical activity in the brain. It consumes energy in the form of "ATP" (and CTP, GTP, TTP...) that came from glucose. All this, in biochemistry, is measured in J/g, cal/g, cal/mol, or similar physical quantities. So, the path is ready for your entreprise. But it would be very complex to achieve the sort of measure you plan (?) to do.

  • Thanks for your contribution. In fact I'm interested in the minimal value that could be reached even for example on a synthetic brain, like a computer...
    – draks ...
    Commented Feb 3, 2014 at 9:42

Creating a proof or theory is a team effort, a discontinuous process occurring over many years. Do you mean the total energy or the just the amount used by the last person of the last stage of the process? The problem is establishing a baseline, knowing when to start measuring and what knowledge, events and other theories were necessary to create the new knowledge. The creator of the theory may have used knowledge discovered years or centuries ago. Do we include the energy of creating that knowledge? Do we include the knowledge that was essential but not directly relevant?

I've ran into a similar problem when I was considering the flow of energy through generations of species, the cost of raising an individual to adulthood, or how much energy is "lost" from the gene pool when prey is eaten by a predator.

I agree with others here that getting an rough, approximate upper bound is definitely feasible by considering how much energy the brain uses and estimating how long someone was working directly on the proof. Finding an exact lower bound may be possible with an AI where you have complete access to the internal state of its mind and could separate what lines of thought were necessary for creating the theory and what was irrelevant. Many hundreds of simulations of the AI creating the proof would have to be run to get an average or minimum, and each run would have to be a completely unique way of obtaining the proof. Even that would not be a "minimum" in the multi-universal sense, over all physically possible ways of obtaining a proof. Doing a calculation involving the bits flipped in a single proof discovery would not give a minimum.

  • 1
    Interesting point about building a proof. Thanks...
    – draks ...
    Commented Feb 3, 2014 at 9:44

No. Because of insight. No. Because "it just came to me." ( said the genius )

The problem with insight as an exception to the premise of your question, is "in a flash" an entire problem or creation can be laid bare in the mind. Others may never reach the point of "productivity" inherent to who another individual "is," and if they did, it might take +1000x the energy expense. There is no level playing field in creation. Some creations are not the product of effort, but expressions of self.

The expectation that energy is a requirement of creation is against the best of the best, and how their works result. And inversely to how eating an Apple burns more calories than it contains, with true creation, more energy is gained than is expended. And I am not reducing energy expenses to brain activity, or physical phenomena by mentioning an apple. I do not believe energy expense in creation and problem solving to be even minimally brain oriented. The brain is not where the work occurs, or where the energy is expended, it is an instrument. Yes, energy is expended there for the tool to function, but that is an entirely different item, completely irrelevant of the root and pertinent activity performed.

No. Because hallucinogens would disrupt the entire basis of the math.

It is a well known fact that the states entered by LSD, peyote, MDMA, DMT, psilocybin, etc... are game changers for those contemplating extremely complicated problems, or pursuing radically creative undertakings. Those same projects could take 100x the time and energy, or be unattainable without conscious expansion. And I'll sidestep the onrush of claims that hallucinogens are alterations and not standard-worthy factors here by saying, "you don't know what you don't know you don't know." If insight is a factor in energy expenditure, so are all other means of reaching a new conscious plane.

No. Because time is relative, beyond the physical plane... and time is flexible.

How fast is the energy expended? In an instant? How long is that instant? How much energy is expended during that interval of immeasurable time? The questions continue on and on, making quantifying energy expenditure impossible to calculate if time is ever a factor in the calculation, because the participant and observer in this experiment are both outside the frame of physical space and time, as much as they are in it. More than the doubt I raise, time is elastic, and the level of enjoyment and engagement modifies the amount expended. The "time flies when you're having fun" concept is not just figurative or subjective, there is a change that cannot be calculated, which alters the kind or amount of time and energy expended. It is on a level that blurs the line between environment and participant, and between participant and observer, and between energy and time, and between space and energy, and between energy and space, etc. "There is no spoon" is a show stopper for quantifying energy expense in creativity or problem solving. There is nothing to measure, statically.

No. Because: Energy is Exogenous. No. Because: Energy is Endogenous.

Depending on your perspective on the creation or problem, the location of the creation and the location of the energy are not within the same system always. This can go on forever, trying to enumerate the location where energy is expended and where creation results, but I will end this partial list by saying that the mind is one level of reality, one location for effort to take place. That draws energy from other levels and there is a rate of exchange there, and as I said above about creation being inverse to eating an Apply, that energy exchange itself might distort your calculation by comparing influxes of energy of an entirely different system with calculation based on a lower or different quality or nature of energy.

And, of computers and computation: Lateral thinking and the counter-part in technology is also a road-block for quantifying what you want to measure, because the above negative answers also apply to programming. The programmer with a deeper or higher or lighter consciousness, whatever you want to call it, crafts a technology which is far more lateral than brute force in its execution and operation. Here again, there is no way to measure what cannot be statically defined, because it has to do with the nature of the creator, not the environmental factors, the tool operation cost, the circumstances, etc. The vast majority of what you want to measure is "behind the veil" of the self, or concealed by the inability to go into transcendence and return with "data." When the ego is lost and the love of the focus ignites, the individual changes into the creator, more than the creation is manifested. That is not costly, that is evolutionary. It is as if the field becomes the place where the creation originated, rather than the originated creation coming into being. The whole loops in upon itself at every point, which is immeasurable. The entire world has changed by the time the calculation is started, and again while it occurs, and again when it finished, and an infinite number of times in between those points.

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