The other approach sees infinity as existing in some philosophical sense. This approach has its roots in Plato's view that the physical world is a dim reflection of some perfect ideal reality. Infinite objects may not exist physically but mathematicians' concept of them is a window into a deeper Platonic reality that is perfect and absolute.
These mathematical approaches parallel approaches to meaning and value which I call absolutist and evolutionary. The absolutist sees ultimate meaning as something that exists most commonly in the form of an all powerful infinite God. The evolutionary sees life and all of a creation as an ever expanding journey with no ultimate or final goal. There is only the journey. There is no destination.
This video argues for an evolutionary view in our sense of meaning and values and in our mathematical understanding. There is a deep connection between the two with profound implications for the evolution of consciousness and human destiny.
The Greek philosopher Zeno argued that:
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. (note 1)Of course mathematicians have long known how to deal with this and for more complex seeming paradoxes of continuous structures, but that does not mean that Zeno was wrong in arguing that an infinitely divisible continuum is impossible.
A modern version of Zeno's paradox exists in our understanding of real numbers. Cantor proved that one cannot map the integers onto the reals. An onto map is a list that pairs every real with a unique integer. Cantor's proof assumes such a map exists and then constructs a real not in the map. The missing real differs in its nth digit from the nth real in the list for every integer n. It is simple to construct such a real.
A counter point to Cantor's proof is the Lowenheim Skolem Theorem. This theorem observed, in effect, that a mathematical system is a computer program for enumerating theorems. All the reals that provably exist in any mathematical system will have their definition enumerated for the first time at a unique finite time step in the execution of this program. Thus every such real can be mapped to the unique integer corresponding to when its definition is first enumerated. No formal mathematical system that humans can construct defines the true set of all reals.
Today some mathematicians think the Continuum Hypothesis is true, others think it is false and still others think it is neither true nor false. Solomon Feferman, the editor of Gödel's collected works, observed:
I am convinced that the Continuum Hypothesis is an inherently vague problem that no new axiom will settle in a convincingly definite way. Moreover, I think the Platonic philosophy of mathematics that is currently claimed to justify set theory and mathematics more generally is thoroughly unsatisfactory and that some other philosophy grounded in inter-subjective human conceptions will have to be sought to explain the apparent objectivity of mathematics. (note 2)
Thirty years ago I told professor Feferman that I thought that the Continuum Hypothesis was neither true nor false in any objective sense. I argued for an evolutionary approach to mathematical truth. This approach defined which mathematical statements were objectively true or false and which are artifacts of the formalization of mathematics.
Many mathematical questions are determined by an infinite sequence of events that one can program a computer to output. Such statements have meaning in an always finite but unbounded universe. They do not require that infinite objects exist. Most mathematical questions of practical significance meet this test (note 3). For example many questions about all real numbers are determined by a finite process operating on every finite initial segment of every real number. One can easily program a computer to enumerate all of these events. This is not true of the Continuum Hypothesis.
Mathematics, like every other human activity, is a product of biological and cultural evolution. The mathematically capable mind evolved because it helps up to control and predict the consequences of our actions. Arithmetic is very useful even in primitive cultures. Abstractions based on arithmetic were essential to the evolution of science and technology. Perhaps abstraction goes too far when it divorces itself from the universe of discrete events that we know exists and speculates about infinite objects that may not exist.
Today there are two dominant physical theories, quantum mechanics and general relativity. Unfortunately, the two theories are incompatible. General Relativity describes what happens at large scales and quantum mechanics describes what happens at small scales. There are no practical experiments where you need both theories. However, at very small time and distance scales, known as the Planck time and the Planck distance, both theories apply. There is no way in existing or foreseeable technology to make measurements at the Planck scale.
That has not stopped physicists from coming up with theories. For the last quarter century particle physicists have focused on a single approach, string theory, for making quantum mechanics and general relativity compatible. Two recent books, The Trouble with Physics and Not Even Wrong lament the apparent failure of this work to come up with a coherent theory or experimental predictions. (note 4).
String theory and most other contemporary approaches to physics rely on continuous mathematics. There have been proposals to consider completely discrete mathematical models. These are models that only need the integers and not the reals. One approach is based on cellular automata proposed by Edward Fredkin and recently promoted by Stephen Wolfram (note 5). I have suggested a closely related idea based on discretized finite difference equations(note 6). These differ from cellular automata in not having a fixed upper limit on the information contained in a single space-time point.
There are two problems with such approaches. First they involve discrete and thus almost inevitably nonlinear effects at the Planck scale. This makes it extremely challenging to come up with predictions that can be measured at the scale of existing experiments.
The other problem is the nonlocal nature of quantum mechanics. In a local theory only what happens in the immediate temporal and spatial neighborhood of a point affects what happens at that point. Quantum mechanics predicts that two events at the opposite ends of the galaxy can have instantaneous mutual influence. The mutual nature of the influence combined with the probabilistic nature of quantum mechanics makes it impossible to send a signal using this effect. The direction in which the effect occurs is encrypted with quantum randomness. Thus predictions remain the same in any relativistic frame of reference and there is not a direct contradiction with relativity.
It is possible to make nonlocal discrete models, but none of the approaches mentioned are nonlocal. A series of experiments have all been consistent with quantum mechanics (note 7). Most physicists consider this to be a dead issue. Some of the more recent experiments involve separations on the order of kilometers and very precise time measurements. However extraordinary predictions require extraordinary verification. No existing experiment has simultaneously closed all major loopholes. I think one needs to be more cautions with such an extreme prediction. Locality is arguably the most powerful simplifying assumption in all of physics. I suspect the physics community has been far too cavalier in abandoning this principle in favor of the elegance of their mathematical models.
The greatest challenge to an absolutist approach to mathematical infinity comes not from questions about the continuum but from Gödel's Incompleteness Theorem. At the dawn of the 20th century, the worlds leading mathematician, David Hilbert, challenged the mathematical community to come up with a formula for deciding all mathematical questions. In the 1930's Kurt Gödel proved this was impossible. He proved that any mathematical system beyond a certain level of complexity contained mathematical questions that could not be decided within the system.
Gödel's proof and the mathematics that has evolved from it are the most powerful evidence supporting an evolutionary approach to mathematical truth. It is impossible for any finite being to know or determine all mathematical truth. This is true even for questions that are logically determined. For example there can be no general finite solution to the question will a computer program ever accept new user input. One can know exactly what the computer will do at any given time but not if it will ultimately do something. The question of whether a computer will ultimately do something is logically determined but not decidable for the general case.
There is only one way around the limitations revealed by Gödel in a finite but perhaps unbounded universe. Instead of pursuing the Platonic ideal of a perfect mathematics one must explore all possible alternatives without any hope of deciding which is correct. Many approaches may ultimately be seen as incorrect but the number of plausible paths must continually increase. This is the process that created the mathematically capable human mind. Biological evolution has followed an immense number of paths in the form of species. This enormous diversity seems to have been essential in evolving the human mathematical mind.
Most mathematicians resist this sort of interpretation of Gödel's result. They see mathematics as the one certain truth in a very uncertain world. There is a core of mathematics that comes close to the ideal of absolute truth. It is a kernel that most mathematicians accept. It is a gift from our evolutionary past. To get seriously beyond it we have to abandon certainty. We need an evolutionary approach to mathematical truth that follows an ever increasing number of alternatives just as biological evolution did in creating us.
The relationship of physical structure to conscious experience is an ancient problem that has yet to be solved. The deeper we look into the structure of the human mind, the more we are able to explain human behavior and the structure of conscious experience. For example we know a great deal about optical illusions from studying the eye and the neural networks that process visual signals.
It appears that in time we will be able to fully explain human behavior and the structure of consciousness from our understanding of the physical structure of the brain and body. We are a long way from doing this but the evidence suggests it will happen. There is no significant scientific evidence pointing in the other direction.
At the same time there is nothing to explain the essence of a conscious experience such as a spectacular sunset. Bertrand Russell over 75 years ago at the end of The Analysis of Matter may have been the first to comment on this disconnect (note 8). Science can only explain structural aspects of our immediate experience. We cannot explain the experience of the color blue to someone who was color blind since birth. We can only communicate structural aspects of blue. For example we can describe a color as pure blue, light blue or reddish blue. This has a scientific interpretation to anyone with the requisite knowledge but it corresponds to a conscious image only for those who have experienced the color blue.
In 1948 Claude Shannon gave a definition of information that makes this distinction clear. He observed that information is communicated when one is able to limit the number of possibilities (note 9). For example if you know a flag will be red, green blue or yellow and you are told it is green then you have reduced the possibilities from 4 to 1. It takes a number between 1 and 4 to select the correct alternative. This definition of communication applies equally to the Internet and to the neural networks that create our conscious experience. The neural nets in our body can communicate a measurable number of selections of possible alternatives in any given time interval. That is their bandwidth a property they share with all communication channels.
So when and how do the neural firings in our body become our conscious experience? If every aspect of consciousness that we can measure or communicate will eventually have a scientific explanation where does the essence of experience, the blueness of blue, fit in? Our scientific and mathematical understanding is devoid of any intrinsic nature. Mathematics is completely abstract building everything from the empty set or nothing at all. Physics has become completely mathematical with no fundamental entities with intrinsic properties like the billiard ball like particles of Newtonian physics. Intrinsic nature exists only in our conscious experience.
Thinking about this forty years ago, I reached the conclusion that consciousness or immediate experience in some form is the essence and totality of the existence of physical structure. The only thing that exists is consciousness. It has a physical structure that we can measure and describe and an intrinsic nature that is beyond description that we continually experience. There is nothing special about the matter that makes up the human body and brain. It is the structure of this matter that makes human consciousness what it is? The universe is the continual transformation of consciousness and nothing but the transformation of consciousness.
The idea that consciousness is universal in all that exists in shared by many spiritual traditions and by contemporary thinkers from the futurist Ray Kurzweil to the mythologist Joseph Campbell (note 10). The idea that consciousness is all that exists is more radical but not unique.
Connecting the assumption that all physical structure is embodied in consciousness to Gödel's Incompleteness Theorem and the mathematics that has evolved from it suggests some extraordinary possibilities. The richness of human consciousness seems to depend in part on the level of abstraction and self reflection that the human mind is capable of. This level is an infinitesimal fragment of what mathematics proves is possible. This suggests that the evolution of richer and deeper conscious experience has no finite limit. Whatever ecstatic wondrous experience, whatever ultimate orgasm any being ever experiences, it is the merest hint of a shadow of what can be and that will always be the case.
In this view the unbounded evolution of consciousness requires ever expanding diversity. It equally requires ever more resources devoted to each viable path. Many of the instinctive and intuitive feelings we have about diversity, individual freedom, power and control may have a foundation in the mathematical requirements for creative evolution. These often conflicting human instincts are both the source of human creativity and at the root of many of the deepest problems we face. As we gain a deeper understanding of the source of these instincts we have a greater chance of focusing and directing them to serve the creative purpose that they evolved to satisfy.
It makes evolutionary sense that we have spiritual instincts that connect us with the creative aspects of evolution. These are critical for long term survival on a dynamic planet. Spirituality often speaks of other planes of existence. I think these our different time scales.
We are first and foremost a product of several billion years of evolution. That process had far more to do with making each of us who we are than our individual life experience or even the particular pattern of genes that we inherited from our parents. Thus we have both an ego identity as an individual and a spiritual identity as part of an evolutionary process that is far bigger than any individual. We have evolved essential instincts to connects us with both.
Our dominant instincts often center around the ego because the process of evolution can continue only through the preservation of individuals. However, in the long term, the evolutionary process can only continue and thrive through diversity and creativity. Our spiritual instincts are, at critical times, more powerful and connected with richer more ecstatic experience than ego associated instincts. Spiritual instincts dominate the course of our lives as they come strongly into play for life altering decisions. Those who resist by following programmatic paths set by themselves or others often find their lives empty and meaningless. Joseph Campbell said it most succinctly: "Follow your bliss."
Conscious experience is its own meaning and value. There is intrinsic value negative or positive in each moment of experience of every sentient being. The Spiritual quest is the search for deeper richer more wondrous experience. That is an open ended unbounded quest that requires ever expanding diversity.
Many philosophical and religious traditions treat questions of meaning and value the way most mathematicians treat the fundamental laws of mathematics. They are seen as absolutes that are revealed or must be discovered. There are kernels of mathematical and spiritual truth inherent in the evolutionary level we have reached but they are an infinitesimal fragment of what is possible. Further progress requires ever expanding diversity with no way of ultimately determining the true path.
We are at a unique tipping point in evolutionary history. Our science and technology, through the objectivity of experiments, are expanding at an ever accelerating pace. They are providing us with unprecedented wealth and power including the power to destroy humanity. Our sense of meaning and value continues to develop haphazardly with no objective guide. The result is an extremely dangerous disconnect. We lack the wisdom and motivation to properly use the enormous power our technology is producing. Identifying physical structure with conscious experience connects our scientific and mathematical understanding to questions of meaning and value. It can be a starting point for an objective spirituality that has the potential to develop commensurate with our technology and power.
Spirituality is mysterious because it deals with issues that are beyond our current intellectual capacity. It may seem that any attempt to objectify spirituality can only lead to its destruction. This is true of attempts to contain it intellectually. It cannot be contained because it points to an unbounded range of possibilities.
In the worlds of science and mathematics there is nearly universal agreement about a vast body of knowledge that has enormous practical value. Of course there is no agreement about how to extend mathematics and science, but there is agreement on the process that must be followed to bring new ideas into the widely accepted core. In contrast religion and spirituality is filled with arbitrary dogma and often violent disagreement. The power technology is giving the human race makes this increasingly dangerous and ultimately unacceptable. If it continues, the human species will almost certainly destroy itself.
It is possible to develop a core spirituality that has the objectivity of science. Like mathematics, an objective spirituality must be open ended with the potential for unbounded creativity. This is where most approaches to spirituality in our intellectually dominated age run into trouble. Intellect is a very powerful but very limited tool. It likes to constrain things in a universe that it can understand. That is one reason most mathematicians including Gödel himself resist the creative implications of Gödel's Incompleteness Theorem.
The universe is the creative evolution of consciousness. This is beyond understanding and imagination. But we can understand the structure of this process. We can understand how to feed it and what will kill it. We can begin to convert some spiritual intuitions into intellectual understanding. Doing so is essential to focus our spiritual motivations into a framework that will allow the continued evolution of consciousness not to mention the survival of humanity.