We stand at a precipice of human civilization. The gap between our scientific knowledge and our religions is enormous.

On the one hand, we live in a world of monotheism (dominant in the 'West') and pan-theism (dominant in the 'East').

On the other hand, interspersed throughout the world is a growing body of scientifically literate people who peer into the depths of the unknown, struggling to reconcile the knowledge of physics, chemistry, biology and sociology with the underlying traditions of the culture that they came from.

Their knowledge of physics, chemistry, biology and sociology gives no meaning to their existence, and at times, gives an opposite meaning to their existence than what their society and bodies are telling them to do.

We need to create a new religion that can help us cultivate consciousness during the peak of prime, and help us release that consciousness during the end of its life. The tenets of this religion can not be at odds with the growing body of scientific knowledge, and can not be at odds with new knowledge that we have yet to invent/discover.

The underlying symbols of the old traditions may help in this goal, but some of them may hinder the goal mentioned above (cultivate ego, and then dissolve ego). We will have to pick and choose wisely. But decide we must because we need to address the widening gulf between our religion and our science.

For example, science seems to be telling us the following:

That life is a means of bringing the universe to its final state as soon as possible.

Let me rephrase this a few times so as to clarify.

Life is a means of increasing the entropy of the universe. For example, the production of entropy on Earth is greater than if there had been no life on Earth. But by increasing the entropy of the universe, we are speeding up the eventual fate of the universe, which appears to be a uniform state of equilibrium. At equilibrium, there is no entropy production...there is no life. There is just equilibrium (now, of course, this depends on whether there is enough mass in the universe to cause the universe to contract backwards...but either way, the end fate of the universe appears to be one of equilibrium, and of no life.)

The purpose of life is to bring the universe to equilibrium as fast as possible.

While there are a few religions in which this 'purpose of life' may be compatible, the 'purpose of life' mentioned above is quite incompatible with the Muslim-Jewish-Christian religion of the 'West'. In these religions, there is a heaven and everlasting life for the individual. Science does not give us a 'heaven' and suggests not only that life for the individual must end, but also that all life must eventually end.

How do we reconcile life as a means to heaven and life as a means of bringing life to an end? (i.e life as a means of ending all life and eliminating all gradients across the universe.)

I do not know how to reconcile this problem. My suggestion, however, is not to ignore this problem. It seems like the religion of the future will need to be some combination of Star Wars and Hinduism.

The Star Wars element is required in order for us to understand that our goal to populate the universe with life. We must expand to other planets so that we can create life on other planets. We must explore other planets and learn to co-exist with other lifeforms so that together we can increase the entropy production of the universe.

We need the Hindu element in order to understand that universes, life and our notions of gods (even Brahman itself) come and go just like a lotus flower dies and comes back again each year. At equilibrium, there is no male and female. There is no good and evil once we reach equilibrium because all forms and structures dissolve away. But perhaps, the universe will start over again. Perhaps, there are multiple universes. Perhaps, metaphorically, the sleeping Brahman will awake again on a new lotus flower and open his eyes.

What we need is a new Luke Skywalker in the form of Bill Gates and Chuck Bartowski (from the TV show Chuck.) We need a new Leia Organa Solo in the form of Meg Whitman and The Bride (Uma Thurman in Kill Bill). Our heroes are action figures like Chuck or The Bride. But we also need business heroes: we need heroes that build businesses like Bill Gates and Meg Whitman in our TV shows and movies.

We all need to fight, but the question is what are we fighting for. To defeat the Galactic Empire, to revenge an assassination attempt, to defeat crime organizations and protect the US government???

No, what we are fighting for is the capability of growing life on other planets so that the entropy of the universe increases even faster, so that life expands. So that life evolves even more. So that life becomes more complex, more dynamic, and more self-aware. More feedback, more mirrors back on itself, higher levels of consciousness, and larger societies on more planets with more entropy production.

Business, nations, family are routes to speed up the production of entropy, but there is no ultimate right business, nation or family. There is no one right meme.

There is no perfect solution to the question: how to bring the universe to equilibrium the fastest. Some memes are better at this than others, but there is no way to prove that one meme is better than another.

So how do we choose between differing memes? How do we choose between Christianity and some new religion? (perhaps something like what I described above.) What if this new religion incorporated into it the idea that there is no right religion? Why should we believe something that states that it's not correct?

Ultimately, this new religion must be capable of growing faster than all other religions and perhaps must also be flexible enough to incorporate old religions into it smoothly.

What we need is a reconciling of Luke Skywalker, The Bride, Leopold Bloom, Molly Bloom, Bill Gates, Yoda, Meg Whitman, Jeff Bezos, Leia Organa Solo, and The Dude.

We need to avoid the creation of billions of George Costanza's: fearful of death and the ending of the individual ego, afraid of his parents, incapable of owning a business, never to marry and bring new life into world. George Costanza represents what goes wrong when religion and science are divorced from each other. He lives in a world in which he understands neither science nor religion and can gain nothing from either. He lives in a world in which he can't make decisions on whom to date or what to eat because each action carries with too much significance because he doesn't see the purpose of life.

The purpose of life is to increase the entropy of the universe. The question is: are we capable of living up to that purpose, or do we run and hide from it? Are we afraid to give up this individual ego that has sprouted up from the elements when the time arises? Are we willing to live with the individual ego during the peak of our life in order to take advantage of its capability to solve complex problems? (i.e. are we willing to not become Zen Masters until we get close to ego's end? Are we willing to avoid the pitfalls of the twenty year old 'Zen Master' and the pitfalls of the 70 year old George Costanza?)

## Friday, December 31, 2010

## Sunday, December 12, 2010

### Self-Replicating Solar Robots

I've wanted to write about self-replicating solar robots for awhile.

I've been interested in the idea of sending self-replicating solar robots to the Moon for awhile, but just recently, I read an article on this topic by professor Klaus Lackner. He calls his self-replicating robots, auxons.

While the idea has been around for awhile, it looks like Klaus Lackner and his co-authors (Darryl Butt and Christopher Wendt) have done the best job of thinking through all of the chemical reaction that must occur to derive the materials needed produce self-replicating robots.

A self-replicating robot has to collect enough electricity from sunlight in order to be able to build a duplicate of itself. I've defined the work return on investment as the ratio of the total net work (electrical work in this case) generated over the lifetime of the machine divided by the total exergy (electricity also in this case) to build the machine. A self-replicating solar robot requires a 'work' return on investment greater than one, and for the robot colony to grow the return on investment must be much greater than one. Also important is the rate of return on investment, which is related to the net electricity generated per unit time divided by the upfront electricity consumed in building the solar auxon. (The "net" in the numerator means the gross electricity produced minus reoccurring electricity expenses associated perhaps with labor, maintenance, fuel costs. For the solar auxons, the reoccurring work is the electricity required to move, repair, and clean.) The rate of return is typical given in units of %/yr. You can invert the rate of return to approximately calculate the pay-back time and then double that value to find the time required to double the population of the solar robots.

Lackner and his co-authors found that the solar colony could double in size every few months. Though, their numbers here seem to rather optimistic because the payback time for solar PV panels is on the order of magnitude of 10 years, and that's using PV solar cells chemistry that probably consumes less electricity than the cell chemistry that would have to be used by the self-replicating robots. On the other hand, the doubling rate for blue-green algae is on the order of 20 hours. I have yet to check their numbers, so I am just speculating right now. (I'd like to do a full analysis, perhaps as a class project in a future course I teach.)

I'm particularly interested in the self-replicating solar robots because they seem to be the best way of populating the Moon. (Unlike Mars, it seems unlikely that water-carbon-based lifeforms, such as algae, could survive on the Moon. We could probably populate Mars with the introduction of the strong greenhouse gases [to melt the ice caps] and some algae from the Earth.) If the Moon could be covered with solar robots, we could possibly use the Moon as a staging ground for further exploration of the solar system.

The chemical composition of the Moon (depending on location) is roughly 45% silica (SiO2), 20% alumina (Al2O3) and 10% iron oxide (FeO). The metals in these three materials would make up the main components of the self-replicating robots (Si, Al, & Fe). As Lackner found, one would have to develop innovative chemical processing techniques in order to make the Si, Al and Fe from the oxides.

While it's possible to produce silicon from some electro-chemical reactions, the robots could also use electricity to run a high temperature plasma arc that heats the silicon dioxide to 4000 K, at which point the oxygen is released in the moon's atmosphere. Unfortunately, the oxygen won't stay up there that long because the moon can't hold on to its gases as long as the Earth can. (I calculated that the probability of an oxygen molecules escaping from the Moon's gravity is 424 million times larger than the probability of an oxygen molecules escaping from the Earth's gravity.) Although, it still may be possible to build up a sizable pressure of O2 in the atmosphere once the self-replicating robots start to cover the entire Moon.

Once the solar robots populate the Moon, the electricity they generate could be used to produce hydrogen from water so that we could fuel hydrogen rockets for further travels into the solar system.

My question is: would we consider self-replicating solar robot colonies to be living organisms? I think that self-replicating solar robots fit the definition of life, but I admit that it'd be a primitive form of life unless they could modify their computer program, i.e. their "DNA".

What would be interesting would be to compare the size of the computer program required to be encoded into each robot vs. the size of DNA for blue-green algae. My guess is the sizes of each would be comparable.

In the future, I plan to calculate the return on investment and the rate of return of the self-replicating solar robots in order to estimate how quickly they could cover the entire surface of the Moon.

A back of the envelope calculation of return on investment can be made from: 1) the exergy requirement for producing Si, Al and Fe; 2) the collection efficiency of each robot; 3) the life time that each robot lives; and 4) the amount of Si, Al, Fe required for each robot.

I've been interested in the idea of sending self-replicating solar robots to the Moon for awhile, but just recently, I read an article on this topic by professor Klaus Lackner. He calls his self-replicating robots, auxons.

While the idea has been around for awhile, it looks like Klaus Lackner and his co-authors (Darryl Butt and Christopher Wendt) have done the best job of thinking through all of the chemical reaction that must occur to derive the materials needed produce self-replicating robots.

A self-replicating robot has to collect enough electricity from sunlight in order to be able to build a duplicate of itself. I've defined the work return on investment as the ratio of the total net work (electrical work in this case) generated over the lifetime of the machine divided by the total exergy (electricity also in this case) to build the machine. A self-replicating solar robot requires a 'work' return on investment greater than one, and for the robot colony to grow the return on investment must be much greater than one. Also important is the rate of return on investment, which is related to the net electricity generated per unit time divided by the upfront electricity consumed in building the solar auxon. (The "net" in the numerator means the gross electricity produced minus reoccurring electricity expenses associated perhaps with labor, maintenance, fuel costs. For the solar auxons, the reoccurring work is the electricity required to move, repair, and clean.) The rate of return is typical given in units of %/yr. You can invert the rate of return to approximately calculate the pay-back time and then double that value to find the time required to double the population of the solar robots.

Lackner and his co-authors found that the solar colony could double in size every few months. Though, their numbers here seem to rather optimistic because the payback time for solar PV panels is on the order of magnitude of 10 years, and that's using PV solar cells chemistry that probably consumes less electricity than the cell chemistry that would have to be used by the self-replicating robots. On the other hand, the doubling rate for blue-green algae is on the order of 20 hours. I have yet to check their numbers, so I am just speculating right now. (I'd like to do a full analysis, perhaps as a class project in a future course I teach.)

I'm particularly interested in the self-replicating solar robots because they seem to be the best way of populating the Moon. (Unlike Mars, it seems unlikely that water-carbon-based lifeforms, such as algae, could survive on the Moon. We could probably populate Mars with the introduction of the strong greenhouse gases [to melt the ice caps] and some algae from the Earth.) If the Moon could be covered with solar robots, we could possibly use the Moon as a staging ground for further exploration of the solar system.

The chemical composition of the Moon (depending on location) is roughly 45% silica (SiO2), 20% alumina (Al2O3) and 10% iron oxide (FeO). The metals in these three materials would make up the main components of the self-replicating robots (Si, Al, & Fe). As Lackner found, one would have to develop innovative chemical processing techniques in order to make the Si, Al and Fe from the oxides.

While it's possible to produce silicon from some electro-chemical reactions, the robots could also use electricity to run a high temperature plasma arc that heats the silicon dioxide to 4000 K, at which point the oxygen is released in the moon's atmosphere. Unfortunately, the oxygen won't stay up there that long because the moon can't hold on to its gases as long as the Earth can. (I calculated that the probability of an oxygen molecules escaping from the Moon's gravity is 424 million times larger than the probability of an oxygen molecules escaping from the Earth's gravity.) Although, it still may be possible to build up a sizable pressure of O2 in the atmosphere once the self-replicating robots start to cover the entire Moon.

Once the solar robots populate the Moon, the electricity they generate could be used to produce hydrogen from water so that we could fuel hydrogen rockets for further travels into the solar system.

My question is: would we consider self-replicating solar robot colonies to be living organisms? I think that self-replicating solar robots fit the definition of life, but I admit that it'd be a primitive form of life unless they could modify their computer program, i.e. their "DNA".

What would be interesting would be to compare the size of the computer program required to be encoded into each robot vs. the size of DNA for blue-green algae. My guess is the sizes of each would be comparable.

In the future, I plan to calculate the return on investment and the rate of return of the self-replicating solar robots in order to estimate how quickly they could cover the entire surface of the Moon.

A back of the envelope calculation of return on investment can be made from: 1) the exergy requirement for producing Si, Al and Fe; 2) the collection efficiency of each robot; 3) the life time that each robot lives; and 4) the amount of Si, Al, Fe required for each robot.

## Sunday, December 5, 2010

### Pi In The Sky

Pi In The Sky:

A Review of "Where Mathematics Comes From"

I found this book to be quite interesting because it attempts to answer the question: where does our knowledge of abstract concepts in mathematics come from? Their quite unique answer is that abstract concepts in mathematics (i.e. a priori knowledge) can not be learned without a significant amount of sense data (i.e. a posteriori knowledge). I suggest reading the book whenever you have the time; it's well worth your time.

Summary of the Book: Eminent psychologists challenge the long-held conviction that mathematics exists on a transcendent plane above humans

If a tree falls in the woods and nobody is there, does it make a sound? Did the number pi exist before humans studied circles? The question of whether mathematicians discover or create is still a heated topic of debate. Mathematics, in all its transcendent beauty, is assumed to have an objective existence external to human beings. This isn’t the case, according to cognitive psychologists George Lakoff and Rafael Núñez. They believe that mathematics, like all human endeavors, “must be biologically based.” In the book, Where Mathematics Comes From, the authors have the goal of convincing both lay readers and mathematicians alike that cognitive psychology will explain the ways in which we learn mathematics. For them, mathematics exists only as a tool for the human mind to use when perceiving the world.

As simple as this sounds, we should consider ourselves lucky to have such a gift. Their research suggests that only a few animals have the ability to count. Our mind formulates integrals, parabolas and rational numbers by expanding on our innate ability to count. Lakeoff and Núñez argue that mathematics is fundamentally a human enterprise, arising from basic human activities. We can only comprehend mathematical concepts, such as imaginary numbers, through metaphors that compare these concepts to activities in real life.

Lakeoff and Núñez take us on a journey through mathematics’ past and cognitive psychology’s future in an attempt to bring mathematics down to earth. We start in a laboratory watching rats and children in counting experiments and we eventually work our way through the minds of great mathematicians, such as George Boole and Leonard Euler.

In their study of the history of mathematics, the authors observed that each layer of mathematics built upon the ones below. Like the iron and steel that holds together modern skyscrapers, a series of grounding and linking metaphors holds mathematics together. The authors uncovered a few grounding metaphors from which we would not be able to visualize or understand concepts like addition or multiplication. These metaphors are the concrete foundation for mathematics. Linking metaphors connect the different fields of mathematics, such as arithmetic, algebra, geometry, trigonometry, calculus, and complex analysis, just as elevators connect the different stories of a skyscraper.

As cognitive scientists go, Lakeoff and Núñez are as different as they get. George Lakeoff has been studying linguistics and the neural theory of language since the 60s. Rafael Núñez is just beginning a promising career in the field, having devoted the last decade studying the origin of mathematical concepts. What they share in common is a Berkeley Psychology department and an interest in learning where mathematics originates in our mind.

Here is an example of the beauty of mathematics, which happens to be the climax of their book.

e to the i time pi is equal to negative one

e^i*pi = -1

The Euler Equation has remained to this date mysterious, and even somewhat mystical, because it is by no means instinctive to believe that raising an irrational number to an imaginary, irrational power could possible yield negative one! One would expect the result to be imaginary, or at least beyond our comprehension. Lakeoff and Núñez argue that pi, i, and e are more than just numbers. They represent ideas. These ideas can only be grasped by a human mind that understands concepts such as periodicity and rotation.

The authors devote such a large section of their book to the Euler equation because this equation combines many branches of mathematics and therefore is a good place to locate linking metaphors between these branches. To give a flavor for the Euler equation, a few of the metaphors used to understand the numbers i and pi will be presented. The beautiful proof of the equation is left for the authors.

The concept of pi from trigonometry is not the same as the concept of pi that comes from geometry. Pi is no longer just the ratio of the circumference to diameter on a circle nor does it just take on the value of 3.14.159…. It is also a measure of periodicity for recurrent phenomena. To understand pi’s importance to periodicity, image yourself on a circle. Any point will do. Image traveling around the circle. How far do you travel before you return to the same point on the circle? It depends on the radius of the circle, so let’s just say that it’s a circle of length of one unit. You must travel through a distance of two pi before reaching one’s starting point on any circle. In the same way, one must travel an octave to reach the same note on a musical scale. To reach the point opposite from you on the circle, you must travel a distance of pi. Outside of the realm of mathematics, we are left with a language that describes recurrence through the circular metaphor, as can be seen by the line, “I can’t wait for the holiday season to come around again.”

The number i is altogether a mysterious quantity. It is by definition the square root of negative one. If you think about this, i can’t be a positive number. A positive number times another positive number is still positive. However, i times i is negative. By the same logic, i can’t be negative because a negative number times a negative number is positive. So, if it can’t be positive or negative, and it definitely isn’t zero, then it can’t be a real number.

Instead of focusing on the number i, it will be more important to understand multiplication by i. We begin by visualizing the number line. This requires having learned the grounding metaphor that numbers can be represented as points on a line. A real number is one that can be visualized on a number line. From there, we need to add a second dimension, which can be done by drawing an axis perpendicular to the first. For real numbers we have the intuitive understanding that multiplication by negative one means finding a number symmetric with the respect to the origin, which is the same as saying multiplication by negative one equals a rotation by 180 degrees in these 2-D plane. Multiplication by i times i is the same as multiplying by negative one. Two rotations of 90 degrees equal to 180 degrees. So, multiplication by i is the same as rotation by 90 degrees. This means that real number on the first number line move to the second axis when they are multiplied by i.

This explanation of i and pi provides enough background to understand Lakeoff and Núñez’s proof of the Euler Equation using the metaphors found earlier in their book. What they show is that Euler and all mathematicians since who have proved this equation could only have done so by learning important metaphors relating mathematics to physical concepts such as lines, rotation and periodicity.

It is fascinating to wonder whether every theorem and proof in mathematics is written down in one Book, drawn from by a few inspired mathematicians. This seems unlikely though because this would mean that the metaphors we use are universal. A species living on a distinct planet probably does not visualize numbers as points on a line. Perhaps for them numbers are represented as stars in the sky. It could ever be possible for them to have a high level of mathematical sophistication without the use of pi.

A Review of "Where Mathematics Comes From"

I found this book to be quite interesting because it attempts to answer the question: where does our knowledge of abstract concepts in mathematics come from? Their quite unique answer is that abstract concepts in mathematics (i.e. a priori knowledge) can not be learned without a significant amount of sense data (i.e. a posteriori knowledge). I suggest reading the book whenever you have the time; it's well worth your time.

Summary of the Book: Eminent psychologists challenge the long-held conviction that mathematics exists on a transcendent plane above humans

If a tree falls in the woods and nobody is there, does it make a sound? Did the number pi exist before humans studied circles? The question of whether mathematicians discover or create is still a heated topic of debate. Mathematics, in all its transcendent beauty, is assumed to have an objective existence external to human beings. This isn’t the case, according to cognitive psychologists George Lakoff and Rafael Núñez. They believe that mathematics, like all human endeavors, “must be biologically based.” In the book, Where Mathematics Comes From, the authors have the goal of convincing both lay readers and mathematicians alike that cognitive psychology will explain the ways in which we learn mathematics. For them, mathematics exists only as a tool for the human mind to use when perceiving the world.

As simple as this sounds, we should consider ourselves lucky to have such a gift. Their research suggests that only a few animals have the ability to count. Our mind formulates integrals, parabolas and rational numbers by expanding on our innate ability to count. Lakeoff and Núñez argue that mathematics is fundamentally a human enterprise, arising from basic human activities. We can only comprehend mathematical concepts, such as imaginary numbers, through metaphors that compare these concepts to activities in real life.

Lakeoff and Núñez take us on a journey through mathematics’ past and cognitive psychology’s future in an attempt to bring mathematics down to earth. We start in a laboratory watching rats and children in counting experiments and we eventually work our way through the minds of great mathematicians, such as George Boole and Leonard Euler.

In their study of the history of mathematics, the authors observed that each layer of mathematics built upon the ones below. Like the iron and steel that holds together modern skyscrapers, a series of grounding and linking metaphors holds mathematics together. The authors uncovered a few grounding metaphors from which we would not be able to visualize or understand concepts like addition or multiplication. These metaphors are the concrete foundation for mathematics. Linking metaphors connect the different fields of mathematics, such as arithmetic, algebra, geometry, trigonometry, calculus, and complex analysis, just as elevators connect the different stories of a skyscraper.

As cognitive scientists go, Lakeoff and Núñez are as different as they get. George Lakeoff has been studying linguistics and the neural theory of language since the 60s. Rafael Núñez is just beginning a promising career in the field, having devoted the last decade studying the origin of mathematical concepts. What they share in common is a Berkeley Psychology department and an interest in learning where mathematics originates in our mind.

Here is an example of the beauty of mathematics, which happens to be the climax of their book.

e to the i time pi is equal to negative one

e^i*pi = -1

The Euler Equation has remained to this date mysterious, and even somewhat mystical, because it is by no means instinctive to believe that raising an irrational number to an imaginary, irrational power could possible yield negative one! One would expect the result to be imaginary, or at least beyond our comprehension. Lakeoff and Núñez argue that pi, i, and e are more than just numbers. They represent ideas. These ideas can only be grasped by a human mind that understands concepts such as periodicity and rotation.

The authors devote such a large section of their book to the Euler equation because this equation combines many branches of mathematics and therefore is a good place to locate linking metaphors between these branches. To give a flavor for the Euler equation, a few of the metaphors used to understand the numbers i and pi will be presented. The beautiful proof of the equation is left for the authors.

The concept of pi from trigonometry is not the same as the concept of pi that comes from geometry. Pi is no longer just the ratio of the circumference to diameter on a circle nor does it just take on the value of 3.14.159…. It is also a measure of periodicity for recurrent phenomena. To understand pi’s importance to periodicity, image yourself on a circle. Any point will do. Image traveling around the circle. How far do you travel before you return to the same point on the circle? It depends on the radius of the circle, so let’s just say that it’s a circle of length of one unit. You must travel through a distance of two pi before reaching one’s starting point on any circle. In the same way, one must travel an octave to reach the same note on a musical scale. To reach the point opposite from you on the circle, you must travel a distance of pi. Outside of the realm of mathematics, we are left with a language that describes recurrence through the circular metaphor, as can be seen by the line, “I can’t wait for the holiday season to come around again.”

The number i is altogether a mysterious quantity. It is by definition the square root of negative one. If you think about this, i can’t be a positive number. A positive number times another positive number is still positive. However, i times i is negative. By the same logic, i can’t be negative because a negative number times a negative number is positive. So, if it can’t be positive or negative, and it definitely isn’t zero, then it can’t be a real number.

Instead of focusing on the number i, it will be more important to understand multiplication by i. We begin by visualizing the number line. This requires having learned the grounding metaphor that numbers can be represented as points on a line. A real number is one that can be visualized on a number line. From there, we need to add a second dimension, which can be done by drawing an axis perpendicular to the first. For real numbers we have the intuitive understanding that multiplication by negative one means finding a number symmetric with the respect to the origin, which is the same as saying multiplication by negative one equals a rotation by 180 degrees in these 2-D plane. Multiplication by i times i is the same as multiplying by negative one. Two rotations of 90 degrees equal to 180 degrees. So, multiplication by i is the same as rotation by 90 degrees. This means that real number on the first number line move to the second axis when they are multiplied by i.

This explanation of i and pi provides enough background to understand Lakeoff and Núñez’s proof of the Euler Equation using the metaphors found earlier in their book. What they show is that Euler and all mathematicians since who have proved this equation could only have done so by learning important metaphors relating mathematics to physical concepts such as lines, rotation and periodicity.

It is fascinating to wonder whether every theorem and proof in mathematics is written down in one Book, drawn from by a few inspired mathematicians. This seems unlikely though because this would mean that the metaphors we use are universal. A species living on a distinct planet probably does not visualize numbers as points on a line. Perhaps for them numbers are represented as stars in the sky. It could ever be possible for them to have a high level of mathematical sophistication without the use of pi.

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