One, two, three, four - by certain age counting to
hundred, thousand or some other number is a well established
skill. Next, in the first grade, comes addition, and year later -
multiplication. By the end of elementary school we conquer
exponentiation. There is a reason why these operations appear in this
order, and to see it lets make a short mathematical detour. Consider
operation of composition (or stacking up) and look back at addition,
multiplication and exponentiation. In these terms, counting, that is
repeatedly adding one, is the simplest one. In order to compute x+y
one needs to take x as an input and repeat (compose) adding one y
times. To get xy we start with x and repeat addition y times, and
finally for x^y we begin with x and repeat multiplication y
times. This of course can be continued forever leading to even more
complicated operations, but we may not need to go any further. Note
that in this approach the role of x and y are quite different. While x
is an ordinary integer which is an input, y tells us how many times
some operation has to be repeated. A computer scientist would say that
these are different data types. Consequently the obvious fact that x+y
is the same as y+x is not that clear in this setting and the fact that
x and y can be interchanged is no longer true for exponentiation and
beyond. This approach points to a mathematical hierarchy and this in
turn indicates increasing complexity level.
I will try to picture the universe as seen by these three functions. The underlying mathematical hierarchy will imply that these pictures have also increasing complexity and the next one subsumes the previous ones. Each next one will offer a bigger picture and it will provide understanding of the
one below which cannot be accomplished there. This can be viewed as a
mathematical analogy similar to being in dimension one, two, three and so on.
To avoid confusion we will adopt anti-platonistic or phenomenological view of reality which seems to be quite well rooted in the current physical theories. Rather than seeing the reality as a
collection of instantiations of some primordial ideas (ala Plato), we will assume that reality is well defined and quite concrete. However, only small fragment of it is accessible to us and furthermore,
abstract ideas are tokens that we use to communicate our thoughts about it. Consequently, as individuals we can live in quite different worlds, and see the reality quite differently.
Additive World
Additive World is rather flat. It is populated by various quantities and it does not give us significant tools to compare them. Most numbers have units attached to them and it is not easy to sort it
out. We have trillions of dollars of national debt, 10% unemployment, hundreds of microsiverts of radiation from broken Japanese nuclear reactor, and so on. Each number has a song to sing but it all adds up to a lot of noise. In the end, we have only that much capacity for retention of this information, and not much use for it.
Additive World is also static, or at least it does not change in a predictable and quantifiable way. Like everything it evolves and changes slowly. But it carries us through these changes without meaningful control or understanding of these transformations. We can sense the difference and long for "good old days" but the basic fabric of everyday life is more or less constant. It is very possible that Additive World is what was meant for us because it makes us part of nature and guarantees stability.
The main survival tool in the Additive World is memory, and in the evolutionary scale the ability to produce offspring which pays attention to their parent's lessons. With very limited ability to control, predict and craft the future, most energy is used to draw from the catalog of past experiences, individual or group-wise. Having a large repertoire of standard behaviors gives an obvious advantage.
Media and advertising cater to our additive side and throw information at us without requiring or wanting us to process it. Internet, which is heralded as the expansion of human mind, creates a overpowering torrent of data. Citizens of the Additive World are not well equipped to handle it.
Living in the Additive World does not obscure the complexities of the reality. However, complex phenomena cannot be dissected and analyzed at this level. The fact that birds can fly is no different from the fact that Superman can fly - both were gifted to their recipients
without their participation or consent. These and similar things can be acknowledged and taken advantage of but they cannot be fully understood.
On the other hand, the lack of ability to explain things in the Additive World is matched by the lack of ability to ask good questions. In fact in the Additive World which is populated by most
living things there is no cognitive gap at all - they ask no questions and seek no answers. It appears that at the moment humans are the only creatures that can move on to the next level.
Multiplicative World
This is Arcadia of scientific revolution, and the picture of the universe which is responsible for the current state of affairs. Even though Babylonians and Greeks knew quite a bit about multiplication,
the understanding at the level that I have in mind is due to Newton. Calculus embodies the essence of the Multiplicative World. Its logo is a zigzagging function with maxima and minima
clearly mapped out. That is right - a function represents some physical process, placement of maxima and minima carry out nearly complete description of it, and calculus is the technology to find all
this information.
Of course there is much more - Multiplicative World is a world of quantifiable change. In this world plain numbers have meaning. If you divide the anticipated outcome of plan A and plan B
then this number is telling us something and allows to make choices. In the Multiplicative World we can compute all sorts of cool things, build theories and verify them, and visualize the future
before we make it a reality. When our predictive power fails, as it sometimes happen, we can still postmortem the failures to see what we missed.
Concept of probability and randomness makes its debut at this level, together with all sorts of statistical tools. We can quantify uncertainty and deal with situations where too many things get in a way to be precise.
Complicated phenomena are very well explained in this world. Celestial mechanics, theory of electro-magnetic interactions are cases where three laws or four equations yield almost complete understandig of physical reality.
The Multiplicative World is a world of action, it gives a mindset of a conqueror. It is a reality where you shoot first and ask questions later. It is almost certain that it represents an evolutionary blind alley as its main product is extremely efficient yet compartmentalized thinking
subjugated to interest of an individual.
It is not only the technological revolution that has roots in the Multiplicative World. Experimentation with social systems of last 400 years are direct results of it. Marx said "The philosophers have only interpreted the world, in various ways; the point, however, is to change it." This is perhaps the Multiplicative World at its extreme - intellectual arrogance and sense of invincibility that easily leads to mayhem and destruction.
Exponential world
This world is dark, gloomy and not well understood. Very few people live there and I can only get a glimpse of it. The logo of exponential world is a point with three curves coming out of it - a depiction of state of water as a function of pressure and temperature. The curves represent phase transitions between different states e.g. gas vs liquid. This is a much more complicated and general picture than a function with maxima and minima, although it includes it as a very special one-dimensional
case. A phase transition is a more complex change than passing through a maximum or minimum and we do not have good science to describe it. There is a great deal of mathematics that needs to be
developed to deal with these questions and only some of it exists at the moment.
I would say that Darwin was the first citizen of the Exponential World. Theory of evolution unravels phenomena that do not belong to the Multiplicative World and cannot be adequately explained
there. Cosmic inflation proposed by Guth is another such phenomenon. Both explain phenomenally well lots of very fundamental questions yet most people do not understand neither theory. Perhaps macroscopic manifestations of quantum mechanics belong to this level as well. Of these two perhaps only Darwin's theory has a flavor of a law of nature while Guth's theory is just a very plausible way to explain the observable.
We are more or less lost in Exponential World and our power of mapping the future is nearly gone. Not only that, even our ability to reconstruct present from the past - when all the information is given - is not that easy either. The Exponential World is circumscribed by Now and past and future seem rather irrelevant and disconnected. If in the Multiplicative World wisdom is measured by the ability to change things, in Exponential World wise are the ones who make as little change as possible.
The Exponential World is very complex, with many non-linear interactions. There are perhaps many phenomena related to non-commutativity, existence of time probably most important of them. While multiplicative view opened the wealth of possibilities, the exponential world in contrast is much more
negative - most real life processes cannot be predicted accurately andthis is a feature of complexity that appears at this level. The Additive World from the perspective of the Multiplicative World is
easy to understand with all of its inner workings in plain view, and consequently Exponential World ought to be simple from the viewpoint of the next one up. Yet, as I will argue below, our chances of gaining this higher vantage point are practically non-existent.
Phenomena belonging to the Exponential World are present in everyday life, many as artifacts of out of bounds multiplicative thinking. Market crashes, global warming, emergence of new diseases are the well known negatives, but many biological processes such as growth of a cell, phenomena in population dynamics, and so on, are at this level of complexity as well. Finally, human beings as biological cognitive machines may perhaps be completely describable at this level of complexity. The Multiplicative World handles reality extremely well within domains encompassed by various phase transitions, and it only fails when those boundaries are crossed and entirely new phenomena emerge.
The Additive World is a perception that we share with every other species, and it is based on sensing rather than controlling. The Multiplicative world is perhaps uniquely human and it gave us tools to control many aspects of life on Earth. As this seems to be a result of just few mutations - ability to develop language being perhaps the most important one - it is not
clear that our existence is an evolutionary necessity or a fluke that spells doom for us and the rest of life forms cohabiting the planet.
In any case, the Multiplicative World is perhaps the Golden Age of
Humanity where creation of abstract ideas leads to shaping of reality on a global scale. Finally, the Exponential World is the natural final step in the self-discovery as it points out to the inherent limitations.
More worlds
From purely mathematical standpoint it
would be disappointing if there was no higher way of thinking than the
Exponential world, and the hierarchy should go on forever in a
meaningful way. Perhaps it does, yet I am very dubious if human mind
can go beyond the Exponential World, and in fact whether even the
Exponential World can be fully internalized. I am quite convinced that
Super-exponential world would yield god-like understanding of human
affairs and life on Earth, and if so this is probably off limits to
us. This is because humans as species or individuals are specific life
forms that in all likelihood can be fully understood and explained
within the framework of the Exponential World. The sliver of reality
that we carve for ourselves is as complex as we are and thus it puts a
natural boundary on our cognition.
hundred, thousand or some other number is a well established
skill. Next, in the first grade, comes addition, and year later -
multiplication. By the end of elementary school we conquer
exponentiation. There is a reason why these operations appear in this
order, and to see it lets make a short mathematical detour. Consider
operation of composition (or stacking up) and look back at addition,
multiplication and exponentiation. In these terms, counting, that is
repeatedly adding one, is the simplest one. In order to compute x+y
one needs to take x as an input and repeat (compose) adding one y
times. To get xy we start with x and repeat addition y times, and
finally for x^y we begin with x and repeat multiplication y
times. This of course can be continued forever leading to even more
complicated operations, but we may not need to go any further. Note
that in this approach the role of x and y are quite different. While x
is an ordinary integer which is an input, y tells us how many times
some operation has to be repeated. A computer scientist would say that
these are different data types. Consequently the obvious fact that x+y
is the same as y+x is not that clear in this setting and the fact that
x and y can be interchanged is no longer true for exponentiation and
beyond. This approach points to a mathematical hierarchy and this in
turn indicates increasing complexity level.
I will try to picture the universe as seen by these three functions. The underlying mathematical hierarchy will imply that these pictures have also increasing complexity and the next one subsumes the previous ones. Each next one will offer a bigger picture and it will provide understanding of the
one below which cannot be accomplished there. This can be viewed as a
mathematical analogy similar to being in dimension one, two, three and so on.
To avoid confusion we will adopt anti-platonistic or phenomenological view of reality which seems to be quite well rooted in the current physical theories. Rather than seeing the reality as a
collection of instantiations of some primordial ideas (ala Plato), we will assume that reality is well defined and quite concrete. However, only small fragment of it is accessible to us and furthermore,
abstract ideas are tokens that we use to communicate our thoughts about it. Consequently, as individuals we can live in quite different worlds, and see the reality quite differently.
Additive World
Additive World is rather flat. It is populated by various quantities and it does not give us significant tools to compare them. Most numbers have units attached to them and it is not easy to sort it
out. We have trillions of dollars of national debt, 10% unemployment, hundreds of microsiverts of radiation from broken Japanese nuclear reactor, and so on. Each number has a song to sing but it all adds up to a lot of noise. In the end, we have only that much capacity for retention of this information, and not much use for it.
Additive World is also static, or at least it does not change in a predictable and quantifiable way. Like everything it evolves and changes slowly. But it carries us through these changes without meaningful control or understanding of these transformations. We can sense the difference and long for "good old days" but the basic fabric of everyday life is more or less constant. It is very possible that Additive World is what was meant for us because it makes us part of nature and guarantees stability.
The main survival tool in the Additive World is memory, and in the evolutionary scale the ability to produce offspring which pays attention to their parent's lessons. With very limited ability to control, predict and craft the future, most energy is used to draw from the catalog of past experiences, individual or group-wise. Having a large repertoire of standard behaviors gives an obvious advantage.
Media and advertising cater to our additive side and throw information at us without requiring or wanting us to process it. Internet, which is heralded as the expansion of human mind, creates a overpowering torrent of data. Citizens of the Additive World are not well equipped to handle it.
Living in the Additive World does not obscure the complexities of the reality. However, complex phenomena cannot be dissected and analyzed at this level. The fact that birds can fly is no different from the fact that Superman can fly - both were gifted to their recipients
without their participation or consent. These and similar things can be acknowledged and taken advantage of but they cannot be fully understood.
On the other hand, the lack of ability to explain things in the Additive World is matched by the lack of ability to ask good questions. In fact in the Additive World which is populated by most
living things there is no cognitive gap at all - they ask no questions and seek no answers. It appears that at the moment humans are the only creatures that can move on to the next level.
Multiplicative World
This is Arcadia of scientific revolution, and the picture of the universe which is responsible for the current state of affairs. Even though Babylonians and Greeks knew quite a bit about multiplication,
the understanding at the level that I have in mind is due to Newton. Calculus embodies the essence of the Multiplicative World. Its logo is a zigzagging function with maxima and minima
clearly mapped out. That is right - a function represents some physical process, placement of maxima and minima carry out nearly complete description of it, and calculus is the technology to find all
this information.
Of course there is much more - Multiplicative World is a world of quantifiable change. In this world plain numbers have meaning. If you divide the anticipated outcome of plan A and plan B
then this number is telling us something and allows to make choices. In the Multiplicative World we can compute all sorts of cool things, build theories and verify them, and visualize the future
before we make it a reality. When our predictive power fails, as it sometimes happen, we can still postmortem the failures to see what we missed.
Concept of probability and randomness makes its debut at this level, together with all sorts of statistical tools. We can quantify uncertainty and deal with situations where too many things get in a way to be precise.
Complicated phenomena are very well explained in this world. Celestial mechanics, theory of electro-magnetic interactions are cases where three laws or four equations yield almost complete understandig of physical reality.
The Multiplicative World is a world of action, it gives a mindset of a conqueror. It is a reality where you shoot first and ask questions later. It is almost certain that it represents an evolutionary blind alley as its main product is extremely efficient yet compartmentalized thinking
subjugated to interest of an individual.
It is not only the technological revolution that has roots in the Multiplicative World. Experimentation with social systems of last 400 years are direct results of it. Marx said "The philosophers have only interpreted the world, in various ways; the point, however, is to change it." This is perhaps the Multiplicative World at its extreme - intellectual arrogance and sense of invincibility that easily leads to mayhem and destruction.
Exponential world
This world is dark, gloomy and not well understood. Very few people live there and I can only get a glimpse of it. The logo of exponential world is a point with three curves coming out of it - a depiction of state of water as a function of pressure and temperature. The curves represent phase transitions between different states e.g. gas vs liquid. This is a much more complicated and general picture than a function with maxima and minima, although it includes it as a very special one-dimensional
case. A phase transition is a more complex change than passing through a maximum or minimum and we do not have good science to describe it. There is a great deal of mathematics that needs to be
developed to deal with these questions and only some of it exists at the moment.
I would say that Darwin was the first citizen of the Exponential World. Theory of evolution unravels phenomena that do not belong to the Multiplicative World and cannot be adequately explained
there. Cosmic inflation proposed by Guth is another such phenomenon. Both explain phenomenally well lots of very fundamental questions yet most people do not understand neither theory. Perhaps macroscopic manifestations of quantum mechanics belong to this level as well. Of these two perhaps only Darwin's theory has a flavor of a law of nature while Guth's theory is just a very plausible way to explain the observable.
We are more or less lost in Exponential World and our power of mapping the future is nearly gone. Not only that, even our ability to reconstruct present from the past - when all the information is given - is not that easy either. The Exponential World is circumscribed by Now and past and future seem rather irrelevant and disconnected. If in the Multiplicative World wisdom is measured by the ability to change things, in Exponential World wise are the ones who make as little change as possible.
The Exponential World is very complex, with many non-linear interactions. There are perhaps many phenomena related to non-commutativity, existence of time probably most important of them. While multiplicative view opened the wealth of possibilities, the exponential world in contrast is much more
negative - most real life processes cannot be predicted accurately andthis is a feature of complexity that appears at this level. The Additive World from the perspective of the Multiplicative World is
easy to understand with all of its inner workings in plain view, and consequently Exponential World ought to be simple from the viewpoint of the next one up. Yet, as I will argue below, our chances of gaining this higher vantage point are practically non-existent.
Phenomena belonging to the Exponential World are present in everyday life, many as artifacts of out of bounds multiplicative thinking. Market crashes, global warming, emergence of new diseases are the well known negatives, but many biological processes such as growth of a cell, phenomena in population dynamics, and so on, are at this level of complexity as well. Finally, human beings as biological cognitive machines may perhaps be completely describable at this level of complexity. The Multiplicative World handles reality extremely well within domains encompassed by various phase transitions, and it only fails when those boundaries are crossed and entirely new phenomena emerge.
The Additive World is a perception that we share with every other species, and it is based on sensing rather than controlling. The Multiplicative world is perhaps uniquely human and it gave us tools to control many aspects of life on Earth. As this seems to be a result of just few mutations - ability to develop language being perhaps the most important one - it is not
clear that our existence is an evolutionary necessity or a fluke that spells doom for us and the rest of life forms cohabiting the planet.
In any case, the Multiplicative World is perhaps the Golden Age of
Humanity where creation of abstract ideas leads to shaping of reality on a global scale. Finally, the Exponential World is the natural final step in the self-discovery as it points out to the inherent limitations.
More worlds
From purely mathematical standpoint it
would be disappointing if there was no higher way of thinking than the
Exponential world, and the hierarchy should go on forever in a
meaningful way. Perhaps it does, yet I am very dubious if human mind
can go beyond the Exponential World, and in fact whether even the
Exponential World can be fully internalized. I am quite convinced that
Super-exponential world would yield god-like understanding of human
affairs and life on Earth, and if so this is probably off limits to
us. This is because humans as species or individuals are specific life
forms that in all likelihood can be fully understood and explained
within the framework of the Exponential World. The sliver of reality
that we carve for ourselves is as complex as we are and thus it puts a
natural boundary on our cognition.
