I know a professor of theoretical physics, with whom I've had many interesting discussions over the years. (Disclosure: I came to Sufism via science.) I wanted to do an interview on the topics we covered with someone who, like me, had progressed from science to Sufism. For those who are the least bit interested in science, physics, and mathematics, the article below will, I believe, prove quite rewarding. The language is simple and no higher mathematics is involved, except only briefly.
Of the “99 Beautiful Names of God,” one is al-Muhsi (The Reckoner, Appraiser, or Accountant): The One who possesses all quantitative knowledge, who comprehends everything, small or great, who knows the number of every single thing in existence. In Arabic, the root HSY connotes “to number, count, reckon, compute,” “to collect in an aggregate by numbering,” “to register or record something,” “to take an account of something.” I conclude that a more concise rendition in English would be: God the Mathematician.
Of course, another of God’s Beautiful Names, the Omniscient (al-Alim), is all-inclusive, so that God’s Knowledge (ilm) encompasses mathematics, physics, and biology alike. But “the Mathematician” makes it more explicit.
In fact, “quantity” (miqdar) and “destiny” (qadar) both derive from the root QDR, and thus are inseparably intertwined.
On May 18, 2014, I recorded a lively conversation with my friend, who wishes to remain anonymous. Highlights from that discussion follow. Text in bold, in brackets, and below graphics belongs to me.
The incredibly sophisticated nanotech machine designs within a single cell. Watch it and weep. (Go to “Settings” and select 480 for best view.) Then ask yourself: can this be the outcome of any random collocation of atoms? When a cell dies, it has precisely the same components. Why then do they lie motionless in the case of a dead cell?
ORDER AND CHANCE DON’T MIX
So… Where shall we start?
Well… They say, “When a person comes of age, s/he becomes responsible” [religion-wise]. Why? Because a person can comprehend the existence of God by reason alone. The mind is enough to know that God exists.
A flower, a bit of soil, a car. Can these nice things have come about by themselves? We’re talking about initial creation, of course. Once the mechanism is in place, after it becomes self-reproducing, things are easier.
Order, disorder. What I’ve seen in life is, unless it’s cultivated, nothing tends to improvement. If something has a chance of going wrong, it will.
That’s Murphy’s Law.
But there’s such an established order that you don’t have to be a professor, you could be a mountain peasant. When you look around, you see this exists. Your child is born. If you leave it alone, it won’t grow up, the child will die. You have to show it exceptional care. There is no need for intelligence to know that a child has parents. That is, you already know it has parents. And this child that is the universe has a parent too, it has an Owner, a Creator. You go to the moon, you find a color television there. Would anyone in their right mind say, “This TV was formed spontaneously out of the ground”? This is absurd.
But they do say that. It’s called evolution by random mutation.
What do they take refuge in? They take refuge in time. But the law of entropy tells us the exact opposite. Time is more of a negative factor in these matters. Time is something that degenerates, unless there is a driving force supporting the process.
They say that radiation causes the mutations, but in all the examples I know of, radiation has a deleterious effect on living tissue.
Radiation is one of the causes of cancer. “A drowning man will grasp at any straw.” That’s why a child is responsible upon reaching eighteen years of age. Because the child is no longer a child, s/he can analyze and see certain things. As a result, I think the intellect alone is sufficient to comprehend God. Prophethood and so on are something else. They’re more specialized matters.
Now science has a dead-end of this sort. They used to define the law of entropy as: “Left by themselves, systems tend to disorder.” Now they’ve changed this, they’ve removed the word “disorder.” They’re trying to abstract entropy away from disorder, they’re trying not to use the words “entropy” and “disorder” together. This is in the newer textbooks. Because otherwise, you ask: “how did this order come about?” Now they write entropy as an equation, they don’t mention disorder. Physicists have tried to circumvent this, to find a solution to the question of entropy, and have wound up nowhere.
That’s the second law of thermodynamics, isn’t it?
Yes. And a peasant doesn’t call this entropy, but he says, “If you don't tend your garden, you'll get weeds.” If you were to bring together all the ingredients of a cell and shake them up, the probability that something will come of that is inconceivably less than 1 divided by 10130, which is already a vanishingly small number. That is, it’s zero. For all practical purposes, this means zero. [See Appendix A. We’re talking about the first living, self-replicating cell.]
But people usually miss the really important point here. If the probability of something occurring randomly is zero, then the probability that it did not occur by chance is a certainty. 1 - 10-130 = 1 - 0 = 1. Now they don’t emphasize that, of course!
Mind-blowing Animations of Molecular Machines inside Your Body [TED]. To claim that all the intricate mechanisms and processes of life could have arisen from inert matter by blind chance, given no matter how many billions of years, is not just an insult to God’s intelligence, but also to our own. It is to elevate the “intelligence” that can emerge from chance to the level of God’s, to impute the highest IQ to random events. Is that anything other than “chance-olatry”—the worship of chance?
And if you say it will form into a cell if shaken for umpteen billion years, that’s an untestable hypothesis, and hence not science. Actually, quite to the contrary, entropy militates that not long afterwards, you’ll have a homogeneous mixture, and it’ll stay that way. Try it with two or three different powders or differently colored liquids, and you’ll see. Shaking more vigorously, adding more energy, doesn’t change the result.
So time is no solution, either. On the contrary, time has an adverse effect. Hence, a mind that can’t perceive this shouldn’t be considered responsible. Because from the point of view of religion, there’s no responsibility when there’s a problem with the intellect. A sacred verse says, “God casts defilement on those who don’t use their reason” (10:100). So you have to use your intellect. There are so many verses that say “men possessed of minds,” “do you not reflect?” But we use our mind for other things. We know very well how to use it for diabolical stuff.
What do scientists do when they’re desperate? They resort to time. Whereas entropy tells us the exact opposite. So a cause is an unavoidable problem. What do you do to get rid of it? You say there was a big bang, and before the big bang there was something else, and before that… you look for a way to wiggle out. Even if you didn’t know about the big bang, I think one ought to know that this can’t be of itself when one beholds this order. One has to see. This is insight.
[For more on this see the Appendix B, taken from another discussion.]
Interacting Gears Synchronize Propulsive Leg Movements in a Jumping Insect (Science, 13 September 2013; click on picture at right to view animation (size: 3 MB)). Gear technology designed into legs (and hence the genes and DNA) of young planthoppers. The mechanical gear was invented around 300 B.C. by humans. For millions of years, a 3-millimeter long hopping insect known as Issus coleoptratus has had intermeshing gears on its legs with 10 to 12 tapered teeth, each about 80 micrometers (or 80 millionths of a meter) wide. The gears enable the creature to jump straight. The teeth even have filleted curves at the base, a design also used in human-made mechanical gears since it reduces wear over time.
Right: screw-and-nut system in hip joint of the weevil Trigonopterus oblongus. The screw thread is half a millimeter in size. Weevils, of which there are 50 thousand species, are a kind of beetle, and have been around for 100 million years. These are examples of God's handiwork in his aspect of Engineer.
You pose a problem in mathematics. One person sees the solution in a second, another sees it in an hour, a third doesn’t see it at all. I think this is like that, with the difference that psychology plays no role in a mathematical problem. Psychology does have a role when you look at nature and infer God. The way you were raised, what your parents taught you, what you received from your surroundings, can prevent you at that point. Because there’s a phenomenon called hypnotism, and this is a form of hypnosis.
I hypnotize someone, I plant the suggestion: “when you wake up, you won’t see that phone.” After they wake up, I ask for the phone. They just can’t find the phone. These experiments have been performed. And human beings are hypnotized like that, only they’re not aware of it. So that person can’t ever find God, because they’ve been hypnotized since childhood.
They’ve been conditioned.
Conditioning takes time. The Prophet said, “Every child is born a Moslem, their parents turn them into something else.” How do they do that? Just so, by conditioning.
FROM MIND TO MATHEMATICS
So the intellect is very important. But intellect is not enough by itself. Until about the year 1700, we talked trusting our intellect. Science didn’t advance much. We talked for thousands of years. Physics was like history, like geography. Everybody was a physicist.
How did this change? With Newton. Prior to the twentieth century, there are three great scientists: Newton, Galileo, and Maxwell. Maxwell isn’t emphasized that much, but he did something of paramount importance. He’s the one who solidified the mathematization of physics. Newton started the mathematics.
He laid down the “method of fluxions” (differential calculus)…
He introduced mathematics to mechanics. Galileo emphasized the importance of experiment. But Maxwell is the person who wrote down all electromagnetic phenomena in the form of differential equations. So there’s a solid mathematization there. And at that point, a discrepancy in the equations presented itself: a conceptual discrepancy. Maxwell resolved the discrepancy according to his own lights, he balanced the equations by adding another term. That’s when it emerged mathematically that electromagnetic waves exist. And so, we actually owe the foundations of our present technology to Maxwell. The mathematization there is as significant as Newton’s.
The physicists of his time objected. One of the protesters was Faraday. Maxwell mathematized Faraday’s Law, as well. Faraday’s objection at that time was: By itself, mathematics does not include any laws of physics. In other words he’s saying, you’re doing this, but you’re doing it in vain. He objects, he says it won’t contribute much to physics. But Maxwell mathematizes these laws.
Now this is very important in present-day physics. You pose a problem, you build a mathematical model of it. Writing the math is a skill all its own. Maxwell did this, and then the objections ceased. When Newton did it, they said, “You’ve done this, but physics has become a specialized science.” We were all physicists before that. You’ve done this math, but it’s a specialized field. So you’ve reduced it to a very small scope, they said, and the objections continued. The principle of gravitation, for example: you say it’s mathematical, but you don’t explain how it occurs. But after Maxwell, because there was that prediction, the objections ceased.
Hence, mathematics accomplishes a very great thing. Looking at it from the viewpoint of classical mechanics, Galileo says, “Let’s check it with experiment.” The mathematical mind may be beautiful, but it’s not everything. The superiority of the mathematical mind to other kinds of mind is that it is a very concrete form of mind. For instance, there’s water vapor and then there’s ice. But the second is concrete. Water vapor exists, too, but it’s not as tangible as ice.
Now you have your way of thinking, I have mine, she has her own. And the logic of each of us has internal weaknesses which we can’t perceive. But mathematics prevents that. Mathematics has become concrete, that is, it has been tested, formulated, thought through by thousands of people. When you apply mathematics, you’re automatically freed of the weaknesses, the fallacies of your personal logic. So mathematics is a more concrete form of logic, of the mind. I’m saying this in terms of its application to physics. Otherwise, there are fields where it can’t be applied. It can’t be applied that much to psychology. I don’t know to what extent it will prove applicable to neuroscience, to modeling the brain.
But much that is useful has come of this. We know the seven planets, it was thanks to mathematics that the existence of the eighth planet was proved. Mathematics predicts. You do the calculations, they don’t agree. The coordinates don’t match, they diverge. Either our model is wrong, or something else is afoot that we don’t know about. What is required for this to occur? You say, there has to be a planet of this mass in such-and-such a position. They say, look at this point on this day, at this hour, and you’ll see a planet. That’s how the eighth planet was first sighted. Two astronomers, one French and the other British, are involved. Lo and behold, on that day at that hour at that point, a planet [Neptune] is observed.
Now this invalidates Faraday’s claim. He was saying that mathematics could not make physical predictions on its own. What did it do? It predicted. That is, mathematics is usually regarded as a tool. But it’s slowly going beyond being a tool. It’s becoming a means of discovery. It’s becoming something of a trailblazer, a pioneer. A tool is a thing that helps you do something, it’s passed beyond that.
And the same with the ninth planet, too. This time, perturbations in the orbit of the eighth planet led to the discovery of the ninth [Pluto]. But the ninth planet was discovered with more difficulty. And then it was demoted from the status of being a planet. They call them “dwarf planets.” Because of the tenth planet, the ninth was demoted.
Now, back to Maxwell: he says there’s a discrepancy, a mathematical, a logical discrepancy. As he gets rid of that, he finds a wave equation there. Hence he says, electromagnetic waves exist. He calculates their velocity, it turns out to be the speed of light. Therefore, says he, light is an electromagnetic wave. And these are all things that were subsequently verified experimentally. Hertz, Marconi… The basis of today’s technology and communications lies there. This is one of the major breakthroughs.
What did mathematics do? It paved the way for something. It led to a new discovery. After being confirmed by experiment, of course. In physics, one should never forget that principle of Galileo.
Examples of this abound. We now come to quantum mechanics. For instance, in quantum mechanics, Dirac’s equation. Dirac’s equation renders quantum mechanics and relativity compatible with each other. The solutions of this equation are more accurate than those of the Schrödinger equation. But here, too, there is a discrepancy, just as there was in the case of electromagnetics. In order to eliminate the discrepancy, Dirac says, there has to be a particle with the same mass as an electron, but with opposite charge. Within a year or two, the positron is discovered. It was so unexpected that the discoverer was awarded the Nobel Prize. Now what has mathematics done? It has again led to a new discovery, it has served as the means to finding a new physical entity. Again, it has passed beyond being a mere tool. And there are many more examples like this.
THE REASONABLE EFFECTIVENESS OF MATHEMATICS
Now, physicists are amazed by this. Eugene Wigner wrote an article on “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”
Stephen Hawking has a saying like that, too. He asks: “What breathes fire into the equations?”
This confusion arises from the assumption that the system excludes the God concept. Otherwise, they wouldn’t be amazed. Because such precision… everywhere there is a logic, a mind, an Infinite Mind at work. Entropy is the cause of our amazement: how can such order exist? The assumptions are wrong. These phenomena clearly tell us that these things can’t happen by themselves, there is an Infinite Mind here.
And as far as physicists are concerned, this is a real conversation-stopper. From here, scientists and philosophers go on to other things. They say [with mathematician David Hilbert]: “Mathematics is a game.” Well, if it is a game, how come it’s so effective in physics? Mathematics is real. But there is no mathematics in nature.
The numbers 2, 3, … don’t exist as objects in nature.
You infer these yourself. For instance, half-integers. Irrational numbers. Rational numbers. Complex numbers. These are entirely constructs of the mind. For example, complex numbers were invented completely independently of physics, so that certain mathematical equations could have a solution. And what do we find, centuries after they were invented? Without complex numbers, quantum mechanics cannot be formulated. There are four or five formulations of quantum mechanics, all of them require complex numbers. There’s just no way to avoid them.
Isn’t it the same with electricity?
No. Complex numbers provide simplicity there. But you can do the calculations without resorting to complex numbers at all. Here, on the other hand, you can’t do anything without complex numbers. You don’t have that luxury.
And here again, the question arises: Weren’t numbers a construct of the mind? Why are mind and nature such an inseparable whole? These are presumably surprising questions for physicists. Also, there is intellect there, but not every intellect. That’s why Galileo is so important. You have to test it against nature, to check whether that intellect is there or not.
For instance, there are four kinds of what are called “division algebras”: real numbers, complex numbers, quaternions and octonions. If a number has an inverse, it’s part of a division algebra. As you move from the first to the last, you lose a property at each stage. Real numbers have the property of ordering: for instance, 5 is greater than 3. With complex numbers, you can no longer say which is greater, 3 + 5i or 5 - 6i. With quaternions, you lose the property of commutativity, and with octonions, you also lose the property of associativity.
Now real numbers and complex numbers are used in nature, but quaternions and octonions are not. A group of physicists tried to formulate quantum mechanics in terms of quaternions, and nothing came of it. And the same holds for octonions. So that’s why experimentation is so important: you have to check the applicability of your mathematics to reality.
In conclusion, the effectiveness of mathematics is unreasonable only if you exclude God. If you include that concept, then it becomes eminently reasonable.
MATHEMATICAL PLATONISM
Now Plato says that mathematics has a reality independent of us. He says we access it by extensions of the mind, and project it on the physical world. That’s why it’s called a Platonic reality. And the same with love: you love another, that person doesn’t know anything about it, it’s all in the lover’s mind. That’s why that love is Platonic love.
But this Platonic reality is a peculiar kind of reality. Where would physics be without mathematics? We would still be talking. We would be in the situation that existed prior to 1600-1700. There would still be a physics, crude, experimental, somewhat like meteorology. In meteorology you make forecasts. But is it like that now? I launch a rocket, thanks to my calculations I know where it’s going to fall, down to the centimeter. With our calculations, we can predict the exact time and duration of a solar or lunar eclipse that will happen 100 years from now down to the second. Now these are not trivial things. Mathematics equates with the mind, an intelligence that pervades the entire universe.
Now we have trouble admitting this. So we don’t want to see or hear certain things. The question of entropy remains unresolved. The formation of the first living cell remains unresolved. It cannot be resolved, because there’s the law of entropy. Those experiments have been performed, that organic soup has been made. Stanley Miller did one experiment, Sidney Fox did another. You place the gases you imagine composed the atmosphere at that time, you give the electric current, that corresponds to lightning strokes. You get amino acids. Amino acids are the building blocks of proteins, so you conclude that life emerged from there.
But it’s not merely a giant step, it’s an impossible step, from amino acids to proteins, if you're going by chance.
OK, how are these organized? Sidney Fox did that experiment. Nothing came of it. By that time, ten years had passed. And nothing would come of it if they were to remain there for ten million years more, because there’s the law of entropy. We say that given time, we’ll solve this. And that’s just kicking the can down the road.
Now, why is mathematics so effective? Because nature is the product of a mind. There’s an Infinite Mind in the universe, a mind that beggars our minds, that makes us look like mongoloids. Moreover, that Mind also has to possess infinite power, in order to enforce those laws all across the universe, from the macrocosmos down to the microcosmos at every level.
Take a single cell, a single human, a single life form. There’s a phenomenal mechanism there, there’s a monumental set of laws. We’ve understood little bits and pieces of these, that is, what we understand doesn’t amount to much. And that, we understand by isolating. For example, we understand an atom, we try to understand a hydrogen atom.
ATOMISM AND HOLISM
We act from the principle of linear superposition. We dismantle things like a clock and assume that like a clock, they’ll work in the same way when they’re reassembled.
Of course, because our approach is atomistic. We haven’t seen any other kind, we don’t know. And we can’t wrap our minds around it, because it’s nothing comprehensible. Now a holistic approach, that’s something else. It’s the outcome of a different state of consciousness. Since we’re in atomistic states of consciousness, our minds too are atomistic. If we had holistic states of consciousness, perhaps we would have holistic minds. There are people with holistic consciousness. We don’t always understand what they say, because they’re talking from a different state of consciousness. A butterfly has a consciousness of its own, a mind of its own. A human has a consciousness of his own, a mind of his own. It’s like that, that is. There’s a relationship between consciousness and mind.
You always say that “Quantum physics is holistic”…
Not many people realize this. Before Newton, mathematics is at the level of arithmetic. Until quantum mechanics, in classical physics, we understand events atomistically, that is, we understand them one at a time. We draw diagrams, those diagrams have correlates. The resultant of two forces, and so on.
In quantum mechanics, the dose of mathematics is stepped up even more. But our understanding diminishes. We have difficulty in comprehending the phenomena. In classical physics, we thought we understood the phenomena. We could take events on a piecemeal basis. In quantum mechanics, there’s a helium atom, it has 2 electrons and a nucleus, the nucleus has 2 protons and 2 neutrons. But we deal with it as a system. When we speak of the energy level of the helium atom, we don’t mean the energy level of the electron, the nucleus, or the proton, we consider the energy level of the system. The phenomenon is approached as a whole.
What happens then? We can’t draw a diagram. The diagrams we draw are abstract. Hence, they have no pictorial representation. Pictures are out. So, three stages: first, arithmetic. Next, a physics at the level of calculus. Third, again physics at the level of calculus, but depiction is lost. Because our assumptions changed. We approached the phenomenon holistically.
Why did we do that? Not because we wanted to. We were forced to do so. In order to make sense of the experiments. We can’t comprehend the results of experiments. The experiment is there, but its results don’t make any sense. We had to derive this formulation in spite of ourselves. The experiments forced this on us. And what is essential in physics is the experiment.
Then we sat down and thought about what it was we had discovered. We had found something holistic.
How about a definition of “holistic,” while we’re at it?
First, let’s clarify what we mean by “atomistic.” Let’s say there’s an event in the solar system. We take the sun separately, the moon separately, this planet separately. Then we do our calculations. Each component has an identity of its own. The values of every component are important. Now, for example, the helium atom, the hydrogen atom, the individual states of protons, of electrons, are no longer of importance. We’re looking at it as a system, that is, as a whole. That’s what “holistic” is. In other words, not to go from the parts to the whole, but to deal only and directly with the whole.
To make a jump, could we deal with the universe in the same way?
The wave function of the universe. There have been studies like that. [Everett-Wheeler-Graham (EWG), “The Many-Worlds Interpretation of Quantum Mechanics.”] Here’s what this means: let there be a wave function, let all that can be known in the universe be in that wave function. And in the representation of the hydrogen atom, there’s all the information related to the system.
Now this is a significant jump. First, it places us in a more helpless situation. It’s like Gödel’s theorems in mathematics. What do Gödel’s theorems do? They undermine the foundations of mathematics, they make it more insecure. We used to be determinists, we used to know everything. Now, we don’t know everything. We don’t know what we’re going to find when we conduct an experiment. We can only say, you’ll find this with this probability and that with that probability. And I don’t know how correct that is, because in order to say that with certainty, you’d have to conduct an infinite number of experiments. Only the menu I’m offering you is definite. But I can’t tell you which item you’ll discover. Because this is a holistic matter, there’s an indeterminacy there. There’s always this in holistic things: a lack of certainty. We can’t understand it, but in the end, we can know the energy levels. And we can do this with great accuracy. We can observe them in experiments. And this has been a very great success.
[Quantum electrodynamics, or QED, has been tested to an accuracy of one part in 100 billion (more recently, in 2006, eight parts in a trillion). The famous American physicist Richard Feynman compared this degree of accuracy to mathematically calculating the distance between New York and Los Angeles to within a hair's breadth. In other words, this is equivalent to predicting the width of North America with the precision of plus or minus one human hair.]
There’s no such thing in classical physics. But actually, there’s a parallel between classical physics and quantum mechanics. Classical mechanics has four or five different formalisms, quantum mechanics has four or five different formalisms. This is not valid for every formalism. For example, the Poisson bracket formalism of classical mechanics is almost the same as the formalism of quantum mechanics, with one difference. The general appearance of the equations is the same.
To me, this looks like the following: in the Koran, they say Ibn Abbas gave a verse’s hidden meaning by interpreting it differently. That’s not what you understand when you read the verse. And I say, that’s what the equation states, but you have to take it as a commutator. That is, there can be different approaches like that in reading the book of nature. There’s actually a one-to-one correspondence, so you penetrate to a deeper level of meaning.
But you can’t logically prove one from the other. That is, you can’t prove the equations of quantum mechanics starting from the equations of classical mechanics. You see the similarity, but there’s no direct proof.
That sounds like pattern recognition, doesn’t it? That is, there’s a form-al similarity.
It’s not just a morphological similarity. For instance, the values of the commutators are identical. So it’s not only a matter of form. Give me the Poisson bracket of anything, I’ll write down its quantum mechanical equivalent for you. This goes beyond form-al. I know the Poisson bracket of a hydrogen atom, of a harmonic oscillator, I can write down the corresponding equation in quantum mechanics, because of this similarity. And the results are phenomenal. This is a different meaning of “a book with twin verses” [the Koran], that is, they have dual meanings. [The book of the universe is here being compared to the Koran.]
Taking the meaning of “verse” (ayah) as “sign” here…
Of course, not as words, but as God’s universe, God’s signs. That is, there’s a signifier in everything. In fact, there are even deeper meanings, and that happens in quantum field theory. Then you give a slightly different meaning.
Now there are operators, and the things they operate on. If you assume commutation relations in the operated (operand), it becomes quantum field theory, that yields even more accurate results. In other words, there are nested meanings. Maybe that’s the case with everything, I don’t know. I’m saying this in terms of physics. But mathematics has an extraordinary role in our discovery of these.
From the viewpoint of physics, however, not every mathematics is always useful. If the assumptions are valid, if you base your mathematics on those, the result is sensational. If the assumptions are wrong, nothing will come of it even if the math is correct.
That is, mathematics is actually a kind of gardening. Seed, cultivation, result. If the seed is the seed of a thorn, no matter how well you cultivate, you won’t get apples from it. Your seed simply has to be the right seed. And that seed is your assumptions. Why, for example, can’t we reach a result in the case of entropy? We can’t sow the right seed there, due to psychological reasons. That’s our problem. So we continue to be surprised. “The unreasonable effectiveness of mathematics” is not unreasonable at all. Why should you be surprised about the mind of God? [Nothing lies beyond its ken.]
You mean it’s not so hard to pass from science to religion?
You can pass to religion from anything, even from art. Perhaps you’ve heard of the joke: “I used to believe in no God, until I saw her. That’s when my opinion changed.” That is, such beauty can’t be accidental. This art can’t happen of itself. This rose doesn’t grow of itself. This scent doesn’t emerge by itself. This beauty, this intricate design, can’t exist of itself. You don’t have to be a physicist to understand this. Take any phenomenon. After you see the balance, the beauty there, you’ll say, this can’t happen on its own.
Of course, there’s the matter of faith here. Anything can be a cause of faith. But there’s also the verse: “Nobody can have faith unless God desires it” (10:100). Some come to faith easily, others just can’t. But if there has to be an occasion for it, it doesn’t have to be mathematics or physics. But mathematics and physics make it crystal clear. So does medicine. A doctor. If the diagnosis is wrong, you can’t heal no matter what the therapy is, right? But for the diagnosis to be correct, you have to have a firm grasp of the processes. And you have to know that nothing is accidental, you have to know the mechanisms, to be able to reach the right diagnosis.
Feynman explains all this elegantly. There were two objections against Newton: 1. You mathematized physics, you made it specialized. 2. You didn’t explain how gravitation occurs, you called it “action at a distance.” This is magic, and it has to remain so. The sun attracts the earth. How does it do this? The mechanism isn’t described.
This was also Einstein, Podolsky and Rosen’s (EPR) objection to quantum theory. Einstein opposed quantum entanglement on the grounds that it was “spooky action at a distance” (spukhafte Fernwirkungen).
It was everyone’s objection. Einstein turns gravitation into the curvature of spacetime, that has problems of its own. For two hundred years, people tried to devise a mechanism for it. There isn’t any. According to Feynman, there’s no difference between saying that gravitation attracts and that “the angel of gravitation” performs the attraction, because we don’t know what it is.
For example, how does a proton attract an electron? Via an “electric field.” These are just words. Are these empty concepts, or can they be filled with meaning? That’s what we have to look at. Except that in quantum field theory, there’s an exchange of photons. We call them “virtual photons.” This tosses a photon to that, and vice versa. That’s how attraction occurs. Mathematically, many nice things have emerged from this. In the weak interaction (weak nuclear force), there is an exchange of W and Z bosons instead of photons. And in the strong interaction (strong nuclear force), gluons are exchanged. All these are by analogy.
But there’s nothing there. There is no impressive prediction. Those in the know don’t say it out loud, but they know and feel it in their hearts. Because the assumptions are wrong, nothing comes of it. It’s the same in every science. Science is an activity performed by humans, and human beings have egos.
FROM SCIENCE TO GOD
How did you pass from science to religion?
From the intellect rather than from science. But science refines this further. You see the accuracy more clearly. Let’s say that a human with a mind, anyone intelligent enough, can comprehend that all this can’t happen by itself when s/he looks at these relationships, this order, this art. But when you go deeper into the relationships, you discover how finely tuned, how delicate, how highly ordered the relationships are, with such great precision, and that cements it. That’s the real contribution of science.
For instance, a doctor. When a doctor goes into that, s/he begins to see things on more of a micro level. They see much deeper than you or I do. So what happens? That cements it. And the same in other places, as well. For instance, if the distance between the sun and the earth were not what it is, there would be no life on earth. There are a thousand things like that. These things wouldn’t be if the ratio between gravitation and the electromagnetic force were not what it is. You perceive that so many coincidences just can’t coincide by themselves.
I actually found God before science, but science riveted it. For example, human beings, couples. There’s a man, a woman. God created them compatible with each other. From that union, a child is born. He gave affection so that that child could live, He created that environment. The male seed, the female seed, there’s an extraordinary design. At that stage, there’s no need to be a physicist to see this. What’s really important here is the patent. Once the factory is in place and working, things are a bit easier. I built up the shop, I left it to my child and went off. The child’s task is a bit easier. Forming it is more difficult. But if the children can’t take it forward, it’ll degenerate and get closed. “It happened by itself.” If so, why can’t I take it forth? Why can’t the child, who took it over in a ready state, take it forward? Therefore, it didn’t happen of itself.
Now, this logic is all very clear, very simple. But you won’t see it if you don’t want to. That’s the real issue. One has to be blind to not see it. Or you have to have grown up blinded. For me, it’s impossible not to see.
FROM GOD TO RELIGION
Now, it’s possible to pass to the concept of God from science or art or something else. But how do we go from the God concept to religion?
There, a vehicle is needed. The mind sees: OK, there’s something here. Why is experiment important in physics? The mind can’t solve everything. Reason has to be tested against reality. Experience is more important. Sometimes we know something from experience, we construct its reasoning later.
To understand religion completely, experience is very important. The phenomenon of prophethood. You can’t understand that with physics, with mathematics. The phenomenon of sainthood, you can’t understand it with the intellect. Our Prophet inspired such a sense of trust in everyone, but in spite of that, not everyone believed in him. Either that, or you have to be able to reach great conclusions from small experiences you live. You saw something in your dream, the next day it took place, it came true. This happened once, twice, three times, ... There’s no place for this in science. Well then, hold on, friend, there’s something here that eludes your intellect.
Now of course, this gives way to listening, to heeding. Why don’t literate people take religion seriously? Because they trust their own mind and do not listen. They don’t listen, they don’t feel the need to listen. First, they were raised that way. Second, they haven’t had experiences like that to astound them. Even if they have, they feel the immediate need to rationalize it. They bypass it. Otherwise, if only they were to start researching, the place to be reached is clear. There’s a world you don’t know, a whole range of experiences you don’t know about. It’s all here. We call it the world of light. There’s the Realm of Power (Jabarut), the Stage of Nondetermination (La ta’ayyun), right? If you ask when that was, they say it’s all simultaneous. That is, they’re all here, and they’re here according to the level of consciousness you’re in.
You mean they’re not in any temporal sequence.
They’re not. Not anywhere else either, they’re actually here [and now]. That doesn’t mean nobody sees them. And that’s our main error. For example, I study mathematics, but I don’t understand it. That doesn’t mean nobody understands it. Or, there’s going to be an earthquake, a dog hears it, I can’t hear it. In other words, there are things I can’t perceive. For example, elephants can hear a sound from a distance of ten kilometers. Its ears are designed that way. Its trunk is designed to emit that sound. That is, both its transmitter and its receiver are suited to the task. My ears and mouth haven’t been designed for that. So the sizes and frequencies tally. Because its wavelength is greater, its frequency is lower. I would be wrong to claim it doesn’t exist.
This has also been said of vision: of the electromagnetic spectrum, we see only a tiny sliver.
Of course, of course. Now they can photograph the same place in every spectrum.
This is used in science, it’s even used in daily life. A thing that can’t be seen at one frequency can be seen at another. Why didn’t this exist before? It wasn’t done until now because we said, this can’t be. In the infrared spectrum, you see something there that you don’t normally see. So we shouldn’t trust our own perceptions too much, just as we shouldn’t trust our intellect too much.
This is also an ego problem. The stronger your sense of self, the more heedless you are, the more you trust yourself. And the greatest catastrophes occur because of that. It’s also true in daily life: you trust yourself too much, your company folds. And such like.
Either you have to have nonordinary experiences, or you have to have experienced people by your side. They explain certain things to us. But of course, in order to understand these events, holistic concepts are needed. This makes comprehension even more difficult.
Do we need to think holistically in order to understand religion?
Religion [Islam] has its own kind of classical mechanics, that’s the Divine Law. It has its quantum mechanics, that’s Paths and Schools. For example, religion tells us, “Do this and this,” “Don’t do that and that.” These are things at the atomistic level. You have to do them yourself, you’re not exonerated if someone else does them. To understand other concepts, holistic things enter: “He who kills one person, kills entire humankind. He who saves one person, saves entire humankind” (5:32). Or, “Don’t gossip, you'll put that person’s spirit in pain.” You find that you are no longer yourself, everything is interlocked, everything is connected with everything else.
Holistic concepts are less well-understood, more delicate things. One reads them in one way, another in another. Like in classical physics versus quantum physics. The second taxes you from a holistic viewpoint, you understand with difficulty unless you’re used to it in terms of experience. If not, you shouldn’t deny, you shouldn’t take risks. That’s what the great Sufi saint Ibn Arabi says: “Even if you don’t believe, don’t deny.” Don’t say, How can this be? “What is in the universe, that is in man.” Don’t say this is impossible. You don’t have that, but don’t say nobody can have it, don’t take that risk.
Now this is entirely holistic. Everything is in the human being. “In man there’s a mountain,” as the Master said. Well, I see no such thing? I can’t reconcile a mountain with a human being. Neither my intellect nor my spiritual condition are up to the task. I can’t understand quantum mechanics, either. Nothing in a high school student is ready for quantum mechanics. And those who understand aren’t entirely there either, but at least we agree that there’s truth in it. There’s a similar situation here. You can’t explain everything to everyone, because they won’t understand. Plus, maybe there’s nothing to be understood, only something to be experienced.
MIND AND RELIGION
Mind alone is not sufficient to discover religion. The mind that comprehends the existence of God is responsible religiously. In order to go beyond that, you need an extra grace from God. Belief in God is a must. For that, the mind is enough. But believing in religion, believing in the Prophet, is a grace from God. There’s a verse to the effect: Noah says, “I’m telling you these things, but they’re no use if God doesn’t wish it.” As Joseph’s brothers are going on their second visit to him, their father Jacob says, “Enter through separate gates. But if God doesn’t desire it, it won’t work.” No matter what you do, it’ll make no difference.
Now we don’t understand this. We don’t understand the will of God. The Master once said, “God scattered a light. It struck some and didn’t strike others.” We don’t know the reason why. In particular, faith in the Prophet rests with God. That is, it’s a very special grace, believing in him is very difficult. Because when you say “God,” you bow to a superior authority. But the Prophet? “Well, he’s human and so am I.” There, the ego enters at once. “He could only have been an ordinary man. The conditions then were such-and-such, he said this, he administered, he was wild,” in the end there’s nothing there. “There was a clever man,” you say. And with that, you miss a lot. You need a special favor to believe that our Prophet was very special, that he was very different, that he was “a mercy to the worlds.” There’s no other way. Or else, God has to have given you the aptitude to derive great conclusions from small experiences. Then it’s possible.
The Master riveted this. Until the Master, I reached that faith only with difficulty: the Prophet is a prophet. But the Master riveted down that faith in place. Our Prophet is very special.
Now, this is very hard to believe. He is the best locus of manifestation the world has ever known. To believe like that is very difficult. Why is that true? Because all the Names of God were manifested in him. There’s no need for someone else. Why is there no need for another Book? Everything is in it [the Koran], even if we can’t understand this. So it’s not necessary. Whereas with the others, it wasn’t like that.
Now it’s hard to accept it like this for our mind-dominated human beings. The ego is strong. Even at birth, children are princes or queens. Those egos won’t bend when they grow up. Here, you need to bend. You need to believe that God gave a mind-boggling boon to someone other than you. But I’m the king… In the language of his state, he says, “If He were to give it to someone, He’d give it to me, I’m king.”
But God favors some human beings. Now, we look at the Koran. What’s there that’s bad about it? It says: “Do good, don’t do evil, don’t harm your neighbor, don’t charge interest on money, don’t be a burden to others.” It counsels all that is good. It says, “Don’t hurt anyone.”
It also defines what is good. It says “This is good, do it, that is bad, don’t do it.” Otherwise, goodness is a relative thing. Thieves think what they’re doing is good.
And that is like abandoning your mind to mathematics. Before Newton, everyone had intelligence. They still do, but everyone does things according to their own lights.
In science, you receive guidance from mathematics, in religion you receive guidance from the Koran.
You have to have a reference. Otherwise, everyone has their own reference point. Take morality. Everyone’s ethics is good from their own standpoint. Why are saints necessary? They hold a mirror to you. They show you yourself, they make you know yourself. Otherwise, nobody is aware of themselves. The Master shows you your error with extraordinary finesse. These things are entirely beyond the ken of contemporary human beings, even conceptually. They can’t even conceive of them, they can’t even conceive what they’re missing. These university professors, these people who think they’re clever, they don’t even know what they’re missing. Meeting the Master, I regard as God’s grace. There’s no other explanation. That is, the mind is at sea here. Everybody’s smart. Many university professors are more intelligent than I am. So this can’t be solely a matter of intelligence, there’s something else. I’m not smarter than they are just because I was graced with the presence of the Master.
CONSCIOUSNESS
I realized that the world is not as I thought it was. This left me shaken. From that I passed on to other things. I already had faith in God, I believed in the Prophet, too. Scientists need experiences that will stagger them, experiences that will shake their belief that they know everything. That’s the only way. Because these are matters of consciousness. In its essence, religion has to do with consciousness. You have to observe changes in your consciousness. You’ll realize then that things are different. There are different states of consciousness: your present state of consciousness, there’s hypnosis, there are different levels in hypnosis, there’s the consciousness of sleep, there’s dream consciousness, there’s lucid dream consciousness. Each is different than the other. And there are who-knows-what-other states of consciousness that I don’t know about.
Would you define religion as consciousness alteration?
Here’s how I view religion: religion is the process of becoming worthy of God by changing one’s morality. But as you alter your ethics, that has an impact on your consciousness. That’s of secondary importance. Being moral is more important than being in a different state of consciousness. The person whose ethics, whose character traits, are closer to the Prophet’s, that person is the winner. This is the primary criterion that I’ve come to understand in the long run.
Morality is very important. For example, we read in the Koran: “I chose him for Myself.” This is about Moses: “I chose you for Myself.” And the same for Abraham: “God chose Abraham as His friend.” Many of Abraham’s morals, character traits, are recounted in the Koran: “Abraham was of mild-mannered mien.” It also tells what God looks at: “He looks at your heart.” “God loves these, God does not love those,” right? “God does not love misers,” “God loves the generous,” God has given all the codes.
Those things all pertain to morality. It doesn’t say, “God loves those who go to Mars in one leap.” It doesn’t say, “God loves those who do Spacefolding.” Nor does it mean that God doesn’t love those who do Spacefolding, but it’s important only in the second-third-fourth degree. It’s not important if it’s not there. The Koran states very clearly: “God loves these, God does not love those.” If we were to list these, that’s where religion is.
The Master stated that clearly: “God loved human beings very much.” (Teachings of A Perfect Master, p. 56.)
The “Secret of Islam” is Love, nothing else. But if my character remains at the level of a dog or some other animal, how is God going to love me? That is, religion is more a matter of changing one’s state of morality than of changing one’s state of consciousness. The focus is always on ethics.
After the New Age philosophies, this all became: “Let’s change our state of consciousness.” But without a change in one’s state of morality, a permanent change in one’s state of consciousness can’t be obtained. You go up in a helicopter, five minutes later it comes down when it runs out of gas.
For example, let’s get top grades in the exam. How? Let’s cheat. But the means are more important than the ends. To obtain those credentials legitimately. This is actually stated very clearly in the books of great Sufis. For instance, in the “Holy Bestowal” [by Abdulqader Geylani]. Worshipers: worship is very important. Scientists/scholars: knowledge is very important. The wise: the secret and maybe the state of consciousness are very important. But most important of all is the love of God.
Then, the question becomes: “How can we attain that love?” And that’s not possible except by ethics, and that’s a very hard thing to do. If only our ethics were beautified by our saying so, my ethics would have improved long ago. No, that happens by suffering. By suffering hardships. It’s not easy for a rock to become earth. It happens in time, by suffering hardships. It happens by paying careful attention to principles. It happens by paying careful attention to the Prohibited and the Permitted.
Religion is a matter of ethics, a matter of becoming worthy of God by this means. First things first. That’s what God wants. He says, “First fix your ethics, then come to Me.”
Intelligence is also important in these matters. “Who has no mind has no religion.” There’s a Tradition of the Prophet. Someone said: “My friend is highly moral.” The Prophet asked: “How is his intelligence?” “Not that much.” “Then he can’t progress very far.” On the other hand, if you’re not straight inwardly, the more intelligent you are, the more harmful you are.
But the Master posits courtesy. Why? Because courtesy is actually morality. Courtesy is the refined form of morality. If you want the Owner, you have to fix your ethics.
At first, I didn’t understand that. I’m reading the Koran, it says “those who want Paradise,” but it also says “those who want God.” So there is such a concept as desiring God. What is this? It’s in the Koran. So some people desire God more than Paradise. [The Turkish Sufi poet] Yunus Emre said that, and the expression is in the verses of the Koran. But it’s hard to discern it there.
He sang, “I need You and You alone.”
It’s been said, “When God is present, neither heaven nor hell exist,” right? That is something amazing. Because we want to re-establish our severed link with God [re-ligio]. That’s our real quest. Heaven and hell pale in comparison. When you’re dealing with God, everything pales in comparison.
Of course. Compared to infinity, every finite thing is zero.
It’s like this in our lives, too. How so? When our friends come visiting, we prepare a treat. But our friends don’t come for that bounty, they come for a reunion. The reunion is the important thing, not bounties or Paradise. Now suppose that some come for the food. Well, let them! Let no one remain hungry. But the main point is not the bounty.
Paradise is a boon, a wonderful boon. But in the end, it’s a boon. The phenomenon of Union is very different. What’s important for us is Union, just as it is for God.
I see this in Sufi writings. What God desires is Union. God created human beings for Himself. And He said: “Fix your ethics, and come.” Now of course, God’s bestowals are going to be worthy of His glory. But there’s something that will put even “blessings such as no eye has seen and no ear has heard” to shame. That must be what they mean by “the Truth of Certainty.” You reach the highest level of proximity. Beyond “the Knowledge of Certainty” and “the Eye of Certainty.” That’s how we see the Master, he’s at the level of the Truth of Certainty.
We’re going to perform the Prayer, we’re going to Fast. But what does the Master say? “Even if your head doesn’t rise from prostration, it won’t happen without these.”
So it’s a matter of ethics. Actually, this is religion: religion is the task of making yourself worthy of God. Can we achieve that? That’s another matter entirely. But that’s the purpose. We don’t know if we can go to Mars, but that’s our calling: to go to Mars. It’s not a matter of knowledge, of consciousness. You can have those too, but there’s a ranking in terms of importance.
The important thing is to display praiseworthy conduct. A man rescues a kitten from the rain, that night he dreams that the Prophet is stroking his beard. So it pleased him. And what’s pleasing to the Prophet is pleasing to God as well. He couldn’t have dreamed that if he had spent that whole night in worship. Let him worship, by all means, but the thing is beauteous conduct. That is, God’s pleasure, something that pleases Him.
CONCLUSION
Mathematics is important because it represents the mind. Physics plus mathematics proves God’s existence. For it is by mathematics that we best analyze nature. The root of the matter is there. Nothing is accidental. Everything is calculated, programmed, precise. And this is a very clear indicator of God’s existence.
If the seed is right, it will yield results. God attaches great importance to the intellect. If you have no mind, you’re not responsible. Because you can deduce the existence of God based purely on reason. If you accept the Prophet too, that’s awesome. And mathematics is important because it has become a means of discovery. But if your assumptions are wrong, mathematics won’t help you. If they’re correct, unexpected things can emerge from that. The mind, mathematics, and experiment have brought us to a place in three hundred years that we hadn’t been able to reach in the previous three thousand. It’s magnificent.
Great scientists, and Dirac is one of them, have arrived at the point that from now on, we need to study consciousness. We don’t know how to study it yet. The Sufi masters have been studying it for centuries.
So where Dirac ends, the Sufi masters begin.
Dirac arrived at that point. So did [Roger] Penrose. And that’s where everyone will arrive at, sooner or later. That’s the point where the masters enter the loop. And then, you have to understand the importance of religion better. You have to perceive that religion is important, that morality is important, that things are not as you imagine them, that the intellect alone is not sufficient, in order to come to that door.
APPENDIX A
A small protein may typically contain 100 amino acids, each with 20 varieties. For example, the protein histone-4 has a chain of 102 amino acids. The probability of even one small enzyme/protein molecule of 100 amino acids being arranged randomly in a useful (and hence, necessarily specific) sequence would be 1 part in 20100 = ~10130. For comparison, there are ~1080 protons in the entire universe. Even the smallest catalytically active protein molecules of the living cell consist of at least a hundred amino acid residues, and they thus already possess more than 10130 sequence alternatives. Getting a useful configuration of amino acids from the zillions of useless combinations is an exercise in futility. A primitive organism has about the same chance of arising by pure chance as a general textbook of biochemistry has of arising by the random mixing of a sufficient number of letters. And the moment you say that non-chance events are involved, such as the folding and fitting of molecules, you fall outside the field of randomness. You implicitly admit the presence of order.
It appears that some people lack an adequate understanding of either the mathematical law of large numbers, or the physical law of entropy, or both. The law of large numbers (LLN) solidifies the expected probability or improbability of an event. If an event is improbable to begin with, an extremely large number of trials will only certify that improbability.
Actually, the two are linked: “The law of the increase of entropy is guaranteed by the law of large numbers... order is an exception in a world of chance” (Hans Reichenbach, p. 54-55), and the LLN is at the core of the second law of thermodynamics.
It would be unfair to one of the great names in quantum physics, Erwin Schrödinger, if we were to neglect mention here of his monograph, What Is Life? (1944). There, he explicitly associated life with negative entropy, or “negentropy” for short. This also ties in with Information Theory: information is a measure of order, entropy is a measure of disorder, so information is the negative of entropy.
APPENDIX B: THE RANDOMIST HAS NO CLOTHES
The “randomists”—that’s what I call people who try to explain the origin and development of life by random events occurring over eons—claim that there are highly improbable events which nevertheless occur every once in a while. For instance, winning the lottery is a highly improbable event, yet somebody does win the lottery. And getting a royal flush in a card game is an extremely improbable event, yet it does happen every now and then. Starting from such examples, they argue that highly improbable events can become possible, probable, and even actual, given billions of years.
First, I should perhaps clarify that I’m not opposed to evolution as such. There’s the fossil record and all that. Natural selection exists. Mutations are a fact of life. What I’m against is supposing that extremely highly ordered phenomena, such as we witness everywhere in life, can be the outcome of chance events. Order does not arise spontaneously out of disorder.
[To be more explicit: directed evolution is a possibility, random evolution is not. Nature cannot produce blueprints that have not been encoded into it.]
Now like I said, the reason for this can’t be found in logic. Rather, it’s psychological. Those who make this claim, the “randomists” as you’ve called them, are in a hypnotic state that makes you Godproof. They don’t want to see. These people who impute the most important things to chance: observe them and you’ll see, in their own lives they leave nothing to chance. Because deep in their hearts, they know that chance alone won’t get you there.
The lottery is designed so that at least one person will win. And you need not one, but a run of at least a thousand consecutive royal flushes to even begin to approximate the complexity of life processes. You know Murphy’s law. It says: “If anything can go wrong, it will.” This is actually the law of entropy. And you need, not only intelligence, but also will, to counteract this.
Consider a TV set. One component in the wrong place, and the device won’t work. Now put all the components of a TV set in a sack and start shaking. Do you actually expect that after a sufficient number of shakes, they will all fall into the right place and the TV will assemble itself? First you need a plan, a blueprint. For that, intelligence is needed. And then, you need an iron will and constant, diligent supervision at every step of the way, to ensure that the thing actually gets done. Otherwise, it’s hopeless. Without that, everything tends to disorder, as anyone who’s ever accomplished anything knows firsthand.
Let's say you’re a Martian, and you see the Mars Rover moving about doing things. There’s no human being around, there’s nothing around, and yet it’s doing those things. It seems to be doing everything by itself, but it’s not. Someone has built it and is guiding it from millions of miles away. A chick lives and dies, but someone has to have programmed it, to have arranged it that way. We now have pilotless planes, but they were planned and developed over time. It didn’t happen all of a sudden.
That reminds me of what a friend once said about the “infinite monkey theorem,” as it’s called. There’s even a jingle about it, which I can’t resist quoting here:
There once was a brassy baboon
Who used to breathe down a bassoon
He said: “It appears,
in millions of years,
I’m certain to hit on a tune.”
Now this friend was a doctor, and he said this when he was a medical student, when they were studying the intricate workings of the human body. He said: “OK, I’ll accept that a monkey can actually do that, given infinite time. What I cannot accept is that this human body, with its millions of processes going on simultaneously, can ever be the work of chance.”
How do those who deny the lack of randomness do so? They defer to infinite time. Because you can’t test it. Or they invoke higher dimensions. You can’t test that, either. Or they call it a “quantum jump” [punctuated equilibrium]. That is, they throw the issue into untestable territory.
Feynman’s principles here are great. I like his approach. He says no theory is right. For it to be correct, it has to pass an infinite series of experiments. A theory passes an experiment, that means it has passed that experiment, it has not yet been falsified. [The concept of falsifiability was developed by philosopher of science Karl Popper.]
Today, there’s the situation that when a theory doesn’t conform with experimental facts, you go back and mathematically tweak the theory until it does, and hence you remove the possibility of falsifying it. And that’s an illusion.
There’s a couplet by the famous Turkish Sufi poet, Niyazi Misri, that expresses all this in a nutshell:
Nothing is more apparent than God
He is hidden only to the eyeless.