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Werner Heisenberg 1901 - 1976

The following interview with Werner Heisenberg was conducted by David Peat (DP) and Paul Buckley (PB) in the early 1970's for broadcast as part of a CBC radio documentary series entitled Physics and Beyond. The interview was later published in Glimpsing Reality: Ideas in Physics and the Link to Biology, a collection of interviews with leading scientists of the Twentieth Century. The book is available for purchase through Amazon.com.


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While a student of Amold Sommerfeld at Munich in the early 1920s Werner Heisenberg (1901-75) first met the Danish physicist Niels Bohr. He and Bohr went for long hikes in the mountains and discussed the failure of existing theories to account for the new experimental results on the quantum structure of matter. Following these discussions Heisenberg plunged into several months of intensive theoretical research but met with continual frustration. Finally, suffering from a severe attack of hay fever, he retreated to the treeless island of Helgoland. After days spent relaxing and swimming Heisenberg suddenly experienced the giddy sensation of looking down into the heart of nature and conceived the basis of the quantum theory. He took this theory to Bohr at Copenhagen, and for the next few weeks they argued and probed its implications long into the night. The results of these discussions became known as the 'Copenhagen interpretation of quantum theory' and are accepted by most physicists. Aspects of the interpretation include Heisenberg's uncertainty principle and Bohr's principle of complementarity.

Werner Heisenberg
Werner Heisenberg

Heisenberg made other important discoveries in physics, and became one of the most distinguished physicists of the century. He was awarded the Nobel Prize for Physics in 1932. His scientific attitudes reflect a debt to philosophy and in particular his respect for Plato. Some of his thoughts on science and society are recorded in a readable autobiography entitled Physics and Beyond.

In recent years Heisenberg adopted the unpopular position of criticizing research in elementary particle physics and proposing that symmetries and not elementary particles form the fundamental starting-point for a description of the world. Towards the end of this chapter he touches upon this theory and its reception.

Professor Heisenberg was interviewed one sunny morning in his office at the Max Planck Institute in Munich. We began by asking Heisenberg to recall the early days of quantum theory but it became apparent that great men have no desire to live in the past and he was just as eager to talk about the future of physics.

DP Could you reminisce about the time when you arrived at the idea of quantum mechanics?


Click to hear an extract from the interview

At that time, there was general discussion among young physicists about the possible ways to establish a coherent quantum theory, a coherent quantum mechanics. Among the many attempts, the most interesting for me was the attempt of H.A. Kramers to study the dispersion of atoms and, by doing so, to get some information about the amplitudes for the radiation of atoms. In this connection, it occurred to me that in the mathematical scheme these amplitudes behaved like the elements of a mathematical quantity called a matrix. So I tried to apply a mathematical calculus to the experiments of Kramers, and the more general mechanical models of the atom, which later turned out to be matrix mechanics. It so happened at that time I became a bit ill and had to spend a holiday on an island to be free from hay fever. It was there, having good time to think over the questions, that I really came to this scheme of quantum mechanics and tried to develop it in a closed mathematical form.

My first step was to take it to W. Pauli, a good friend of mine, and to discuss it with him, then to Max Born in Göttingen. Actually, Max Born and Pascual Jordan succeeded in giving a much better shape and more elegant form to the mathematical scheme. From the mathematical relations I had written down, they derived the so-called commutation relations. So, through the work of Born and Jordan, and later Paul Dirac, the whole thing developed very quickly into a closed mathematical scheme.

I also went to discuss it with Niels Bohr, but I can't be sure whether this was in July, August, or September of that year [1925].

Half a year later the first papers of E. Schrödinger became known. Schrödinger tried to develop an older idea of Louis de Broglie into a new mathematical scheme, which he called wave mechanics. He was actually able to treat the hydrogen atom on the basis of his wave mechanical scheme and, in the summer of 1926, he was also able to demonstrate that his mathematical scheme and matrix mechanics were actually two equivalent mathematical schemes, that they could be simply translated into each other. After that time, we all felt that this must be the final mathematical form of quantum theory.

DP Had you and Bohr begun the interpretation of this work before Schrödinger's paper came out?

Of course, there was continuous discussion, but only after Schrödinger's paper did we have a new basis for discussion, a new basis for interpreting quantum theory. In the beginning there was strong disagreement between Schrödinger and ourselves, not about the mathematical scheme, but about its interpretation in physical terms. Schrödinger thought that by his work physics could again resume a shape which could well be compared with Maxwell's theory or Newton's mechanics, whereas we felt that this was not possible. Through long discussions between Bohr and Schrödinger in the fall of 1926, it became apparent that Schrödinger's hopes could not be fulfilled, that one needed a new interpretation. Finally, from these discussions, we came to the idea of the uncertainty relations, and the rather abstract interpretation of the theory.

PB Did Schrödinger ever like that interpretation?

He always disliked it. I would even guess that he was not convinced. He probably thought that the interpretation which Bohr and I had found in Copenhagen was correct in so far as it would always give the correct results in experiments; still he didn't like the language we used in connection with the interpretation. Besides Schrödinger, there were also Einstein, M. von Laue, M. Planck, and others who did not like this kind of interpretation. They felt it was too abstract, and too far removed from the older ideas of physics. But, as you know, this interpretation has, at least so far, stood the test of all experiments, whether people like it or not.

PB Einstein never really liked it, even until the day he died, did he?

I saw Einstein in Princeton a few months before his death. We discussed quantum theory through one whole afternoon, but we could not agree on the interpretation. He agreed about the experimental tests of quantum mechanics, but he disliked the interpretation.

DP I felt that at some point there was a slight divergence between your views and Bohr 's, although together you are credited with the Copenhagen interpretation of quantum mechanics.

That is quite true, but the divergence concerned more the method by which the interpretation was found than the interpretation itself. My point of view was that, from the mathematical scheme of quantum mechanics, we had at least a partial interpretation, inasmuch as we can say, for instance, that those eigenvalues which we determine are the energy values of the discrete stationary states, or those amplitudes which we determine are responsible for the intensities of the emitted lines, and so on. I believed it must be possible, by just extending this partial interpretation, to get to a complete interpretation. Following this way of thinking, I came to the uncertainty relations.

Now, Bohr had taken a different starting-point. He had started with the dualism between waves and particles - the waves of Schrödinger and the particles in quantum mechanics - and tried, from this dualism, to introduce the term complementarity, which was sufficiently abstract to meet the situation. At first we both felt there was a real discrepancy between the two interpretations, but later we saw that they were identical. For three or four weeks there was a real difference of opinion between Bohr and myself, but that turned out to be irrelevant.

DP Did this have its origin in your different philosophical approaches?

That may be. Bohr's mind was formed by pragmatism to some extent, I would say. He had lived in England for a longer period and discussed things with British physicists, so he had a pragmatic attitude which all the Anglo-Saxon physicists had. My mind was formed by studying philosophy, Plato and that sort of thing. This gives a different attitude. Bohr was perhaps somewhat surprised that one should finally have a very simple mathematical scheme which could cover the whole field of quantum theory. He would probably have expected that one would never get such a self-consistent mathematical scheme, that one would always be bound to use different concepts for different experiments, and that physics would always remain in that somewhat vague state in which it was at the beginning of the 1920s.

DP In the interpretation you gave at that time, you seemed to imply that there did exist an ideal path and that somehow the act of measuring disturbed the path. This is not quite the same as the interpretation that you hold now, is it?

I will say that for us, that is for Bohr and myself, the most important step was to see that our language is not sufficient to describe the situation. A word such as path is quite understandable in the ordinary realm of physics when we are dealing with stones, or grass, etc., but it is not really understandable when it has to do with electrons. In a cloud chamber, for instance, what we see is not the path of an electron, but, if we are quite honest, only a sequence of water droplets in the chamber. Of course we like to interpret this sequence as a path of the electron, but this interpretation is only possible with restricted use of such words as position and velocity. So the decisive step was to see that all those words we used in classical physics - position, velocity, energy, temperature, etc. - have only a limited range of applicability.

The point is we are bound up with a language, we are hanging in the language. If we want to do physics, we must describe our experiments and the results to other physicists, so that they can be verified or checked by others. At the same time, we know that the words we use to describe the experiments have only a limited range of applicability. That is a fundamental paradox which we have to confront. We cannot avoid it; we have simply to cope with it, to find what is the best thing we can do about it.

DP Would you go so far as to say that the language has actually set a limit to our domain of understanding in quantum mechanics?

I would say that the concepts of classical physics which we necessarily must use to describe our experiments do not apply to the smallest particles, the electrons or the atoms - at least not accurately. They apply perhaps qualitatively, but we do not know what we mean by these words.

Niels Bohr liked to tell the story about the small boy who comes into a shop with two pennies in his hands and asks the shopkeeper for some mixed sweets for the two pennies. The shopkeeper gives him two sweets and says 'You can do the mixing yourself.' This story, of course, is just meant to explain that the word mixing loses its meaning when we have only two objects. In the same sense, such words as position and velocity and temperature lose their meaning when we get down to the smallest particles.

DP The philosopher Ludwig Wittgenstein originally started out by thinking that words were related to facts in the world, then later reversed his position to conclude that the meaning of words lay in their use. Is this reflected in quantum mechanics?

I should first state my own opinion about Wittgenstein's philosophy. I never could do too much with early Wittgenstein and the philosophy of the Tractatus Logico-philosophicus, but I like very much the later ideas of Wittgenstein and his philosophy about language. In the Tractatus, which I thought too narrow, he always thought that words have a well-defined meaning, but I think that is an illusion. Words have no well-defined meaning. We can sometimes by axioms give a precise meaning to words, but still we never know how these precise words correspond to reality, whether they fit reality or not. We cannot help the fundamental situation - that words are meant as a connection between reality and ourselves - but we can never know how well these words or concepts fit reality. This can be seen in Wittgenstein's later work. I always found it strange, when discussing such matters with Bertrand Russell, that he held the opposite view; he liked the early work of Wittgenstein and could do nothing whatsoever with the late work. On these matters we always disagreed, Russell and I.

I would say that Wittgenstein, in view of his later works, would have realized that when we use such words as position or velocity, for atoms, for example, we cannot know how far these terms take us, to what extent they are applicable. By using these words, we learn their limitations.

DP Would it be true to say that quantum mechanics has modified language, and, in turn. language will re-modify the interpretation of quantum mechanics?

There I would not quite agree. In the case of relativity theory, I would agree that physicists have simply modified their language; for instance, they would use the word simultaneous now with respect to certain coordinate systems. In this way they can adapt their language to the mathematical scheme. But in quantum theory this has not happened. Physicists have never really tried to adapt their language, though there have been some theoretical attempts. But it was found that if we wanted to adapt the language to the quantum theoretical mathematical scheme, we would have to change even our Aristotelian logic. That is so disagreeable that nobody wants to do it; it is better to use the words in their limited senses, and when we must go into the details, we just withdraw into the mathematical scheme.

I would hope that philosophers and all scientists will learn from this change which has occurred in quantum theory. We have learned that language is a dangerous instrument to use, and this fact will certainly have its repercussions in other fields, but this is a very long process which will last through many decades I should say.

Even in the old times philosophers realized that language is limited; they have always been skeptical about the unlimited use of language. However, these doubts or difficulties have, perhaps, been enhanced through the present developments in physics. I might mention that most biologists today still use the language and the way of thinking of classical mechanics; that is, they describe their molecules as if the parts of the molecules were just stones or something like that. They have not taken notice of the changes which have occurred in quantum theory. So far as they get along with it, there is nothing to say against it, but I feel that sooner or later, also in biology, one will come to realize that this simple use of pictures, models, and so on will not be quite correct.

PB At what point does the transition occur from the non-path to the path in a biological system? Is a DNA molecule already a classical object, or is a cell a classical object?

There is, of course, not a very well defined boundary; it is a continuous change. When we get to these very small dimensions we must be prepared for limitations. I could not suggest any well-defined point where I have to give up the use of a word. It's like the word mixing in the story; you cannot say 'when I have two things, then I can mix them.' But what if you have five or ten? Can you mix then?

PB It seems to me that there is something very important here about language. We are living beings formed from coherent structures like DNA and we apparently have classical paths and our existence is understandable within this language. But then we can analyse by reducing these complex, coherent wholes to smaller and smaller parts, and is it nor perhaps this process of reduction that is at the root of the paradox?

I would say that the root of the difficulty is the fact that our language is formed from our continuous exchange with the outer world. We are a part of this world, and that we have a language is a primary fact of our life. This language is made so that in daily life we get along with the world, it cannot be made so that, in such extreme situations as atomic physics, or distant stars, it is equally suited. This would be asking too much.

PB Is there a fundamental level of reality?

That is just the point; I do not know what the words fundamental reality mean. They are taken from our daily life situation where they have a good meaning, but when we use such terms we are usually extrapolating from our daily lives into an area very remote from it, where we cannot expect the words to have a meaning. This is perhaps one of the fundamental difficulties of philosophy: that our thinking hangs in the language. Anyway, we are forced to use the words so far as we can; we try to extend their use to the utmost, and then we get into situations in which they have no meaning.

DP In discussing the 'collapse of the wave function' you introduced the notion of potentiality. Would you elaborate on this idea?

The question is: 'What does a wave function actually describe?' In old physics, the mathematical scheme described a system as it was, there in space and time. One could call this an objective description of the system. But in quantum theory the wave function cannot be called a description of an objective system, but rather a description of observational situations. When we have a wave function, we cannot yet know what will happen in an experiment; we must also know the experimental arrangement. When we have the wave function and the experimental arrangement for the special case considered, only then can we make predictions. So, in that sense, I like to call the wave function a description of the potentialities of the system.

DP Then the interaction with the apparatus would be a potentiality coming into actuality?

Yes.

DP May I ask you about the Kantian notion of the 'a priori' an idea which you introduced, in a modified sense, into your discussions of quantum theory.

As I understand the idea of 'a priori,' it stresses the point that our knowledge is not simply empirical, that is, derived from information obtained from the outer world through the senses and changed into data in the content of our brain. Rather, 'a priori' means that experience is only possible when we already have some concepts which are the precondition of experience. Without these concepts (for instance, the concepts of space and time in Kant's philosophy), we would not even be able to speak about experience.

Kant made the point that our experience has two sources: one source is the outer world (that is, the information received by the senses), and the other is the existence of concepts by which we can talk about these experiences. This idea is also borne out in quantum theory.

PB But these concepts are pary of the world also.

Whether they belong to the world, that is hard to say; we can say that they belong to our way of dealing with the world.

PB But we belong to the world, so, in a sense, these activities of ours also belong to the world.

In that sense, yes.

DP You modified the 'a priori' by introducing it as a limited concept, is that true?

Of course, Kant would have taken the 'a priori' as something more absolute than we would do in quantum theory. For instance, Kant would perhaps have said that Euclidean geometry would be a necessary basis for describing the world, while we, after relativity, would say that we need not necessarily use Euclidean geometry; we can use Riemannian geometry, etc. In the same way, causality was taken by Kant as a condition for science. He says that if we cannot conclude from some fact that something must have been before this fact, then we do not know anything, and we cannot make observations, because every observation supposes that there is a causal chain connecting that which we immediately experience to that which has happened. If this causal chain does not exist, then we do not know what we have observed, says Kant. Quantum theory does not agree with this idea, and in fact proves that we can even work in cases where this causal chain does not exist.

DP In a recent theory of yours, is not causality retained, perhaps in a new form?

We have causality in that sense - that in order to influence something, there must be an action from one point to the next point; no action can happen if there is not this connection. But at this point one gets into rather complicated details.

DP But even so, you do have causality predicated on the idea of separation and action, so this again comes back to a philosophical level: what you mean by separation, and by interaction.

We must speak about 'interaction' and 'separation,' that is quite true, and we use the terms as we did in classical theory. But, again, we see limitation. Complete separation of two events may be possible in classical theory; it is not possible in quantum theory. So we use the terms together with the fact of their limitation.

DP What exactly are the criteria for something to be classical?

I would say the criteria are simply that we can get along with these concepts (e.g. 'position,' 'velocity,' 'temperature,' 'energy'), and so long as we get along with them, then we are in the classical domain. But when the concepts are not sufficient, then we must say that we have gone beyond this classical domain.

Every system in physics (forget for the moment about biological systems) is always quantum theoretical, in the sense that we believe that quantum theory gives the correct answers for its behaviour. When we say that it is classical, we mean that we do get the correct or the necessary answers by using classical concepts (at least in that approximation in which we can describe the system by classical concepts). So a system is classical only within certain limits and these limits can be defined.

DP How would you include things like irreversibility?

Thermodynamics is a field which goes beyond Newtonian mechanics, inasmuch as it introduces the idea of thermodynamic equilibrium, or canonical distribution as W. Gibbs has put it. Thermodynamics leaves classical physics and goes into the region of quantum theory, for it speaks about situations of observation; it does not speak about the system as it is, but about the system in a certain state of being observed, namely in the state of temperature equilibrium. If this equilibrium is not obeyed, then we cannot use thermodynamics. So the whole concept of irreversibility is bound up with the concept of thermodynamic equilibrium.

DP And is this ultimately connected with the idea of a classical limit to something? I am thinking of the measurement problem that always seems to be associated with an irreversible process: that we have a definite result for a quantum mechanical system where the quantum mechanics itself doesn't seem to predict a definite result. That is, the idea of a quantum mechanical measurement seems to be tied up with the idea of an irreversible trend.

Yes, to some extent, because on the side of the observer we do use classical concepts. The idea that we do observe something already indicates something irreversible. If we draw a pencil line on a paper, for instance, we have established something which cannot be undone, so to speak. Every observation is irreversible, because we have gained information that cannot be forgotten.

DP To what extent is this related to the symmetry-breaking of the quantum mechanical system where one gets classical observables?

I would not like to connect it with symmetry-breaking; that is going a bit far. We try to describe the observational situation by writing down a wave function for the object and the equipment which is in interaction with this wave function. Just by using classical words for the equipment, we have already made the assumption of irreversibility. Or we make the assumption of statistical behaviour, because the mere use of classical words for this observation on the side of the system makes it impossible to know the total wave function of object and equipment. But we cannot use quantum theory for the equipment in a strict sense, because if we wrote down the wave function for the object and the equipment, we could not use classical words for the equipment, so we would not observe anything. We do observe only when we use classical concepts, and just at this point this hypothesis of disorder, of statistical behaviour, comes in.

DP With regard to something like ferromagnetism, the quantum mechanical system has given rise to a macroscopic ordering. Is it true to say that a quantum mechanical system has actually broken its own symmetry and given rise to a classical variable, without any talk about a measuring apparatus, or anything exterior to the system?

Let us consider a ferromagnet as isolated from the rest of the world for some time, and then ask what the lowest state of the system is. We find, from the quantum mechanical calculations, that the lowest state is one in which the whole system has a very large component of magnetic momentum. If we then ask 'what do we observe when we consider this system?' we see that it is convenient to ascribe the classical variable 'magnetic momentum' to the system. So we can use classical terms to describe this quantum mechanical behaviour. But this is not really a problem of observation, only a problem of how the lowest state of the system is defined.

PB How does quantum mechanics deal with time flow or does it in fact say anything at all about it?

I would have to repeat what C. von Weizsäcker said in his papers: that time is the precondition of quantum mechanics, because we want to go from one experiment to another, that is from one time to another. But this is too complicated to go into in detail. I would simply say that the concept of time is really a precondition of quantum theory.

PB In the domain where quantum mechanics operates, all of the equations are reversible with respect to time, except for one experiment I believe. So time has more to do with macroscopic classical systems than microscopic quantum systems.

I would say that irreversibility of time has to do with this other system, with those problems which I. Prigogine describes in his papers, and is certainly extremely important for the macroscopic application of quantum theory, and also for biology, of course.

DP Can we talk about a new theory of yours, the non-linear theory of elementary particles? Are you ultimately going to introduce things like gravitation into this theory, and go over to a picture in which space and time emerge?

Again, we have a similar situation as in ferromagnetism. We try to solve the quantum mechanical, or quantum theoretical equation, but we can see that the system acquires properties which then can be described by classical language (e.g. like speaking of a magnetic momentum, etc.). We are hoping that such phenomena as electromagnetic radiation and gravitation also can come out of the theory of elementary particles, and we have reasons to believe that this is so.

DP The idea of symmetry is a very important part of your theory.

Let's begin more simply by speaking about quantum mechanics, disregarding now the difficulties of elementary particle physics. In quantum mechanics we see that macroscopic bodies have very complicated properties, complicated shapes and chemical behaviour and so on. Coming down to smaller and smaller particles, we finally come to objects which are really very much simpler, for example the stationary states of a hydrogen atom. We describe its properties by saying that these states are a representation of the fundamental symmetries, such as rotation in space. So when we describe a system by writing down a few quantum numbers (in hydrogen atoms, we have the principal quantum number and the angular momentum number) this means that we know nothing except to say that this object is a representation of symmetries. The quantum numbers tell us which kind of symmetries we mean; the numbers themselves say that this object has these special properties. Thus, when we come to the smallest objects in the world, we characterize them in quantum mechanics just by their symmetry, or as representations of symmetries, and not by specifying properties such as shape or size.

DP There are symmetries that are nor related to operations in the world, e.g. the internal symmetries such as isospin. What meaning do they have? Do you think they are related ultimately to the properties of space and time?

I suspect that isospin is a symmetry similar to space and time. I cannot say that it is related to them. I would say that there are a number of fundamental symmetries in this world which may in future be reduced to something still simpler, but so far we must take them as given, as a result of our experiments. One of the most fundamental symmetries is the symmetry of the Lorentz group, that is space and time, and then isospin groups, scale groups, and so on. So there are a number of groups which are fundamental in the sense that in describing the smallest particles we refer to their behaviour and transformations.

The idea is that one can distinguish between a natural law, a fundamental law, which determines for instance a spectrum of elementary particles, and the general behaviour of the cosmos, which is perhaps something not at once given through this law. I might remind you, for instance, of Einstein' s equations of gravitation. Einstein wrote down his field equations and thought that gravitational fields are always determined by them. But the cosmos is not unambiguously determined by these field equations, although there are several models of the cosmos which are compatible with them. In the same sense, I would say that there is an underlying natural law which determines the spectrum of elementary particles, but the shape of the cosmos is not unambiguously determined by this law. Logically, it would be possible to have various types of cosmos which are in agreement with it. However, if a certain cosmological model has been 'chosen,' then this model, of course, has some consequences for the spectrum of elementary particles.

DP Are you saying that there exist laws which are independent or outside the universe, outside the world, which reality breaks, or that it breaks the symmetry represented by the laws?

'Laws' just means that some fundamental symmetries are inherent either in nature or in our observation of nature. You may know about the attempts of Weizsäcker, who tried to derive the laws simply from logic. We have to use language to arrive at conclusions, to study alternatives, and he questions whether from the alternatives alone we can arrive at these symmetries. I don't know whether his attempts are successful or not. In physics, we can only work with the assumption that we have natural laws. If we have no natural laws, then anything can happen, and we can only describe what we see, and that's all.

DP Another feature of your theory which seems to go against the current trend- partons and quarks, etc. - is that you feel that no particle is any more elementary than any other.

Even if quarks should be found (and I do not believe that they will be), they will not be more elementary than other particles, since a quark could be considered as consisting of two quarks and one anti-quark, and so on. I think we have learned from experiments that by getting to smaller and smaller units, we do not come to fundamental units, or indivisible units, but we do come to a point where division has no meaning. This is a result of the experiments of the last twenty years, and I am afraid that some physicists simply ignore this experimental fact.

DP So it would seem that elementary particles are just representations of symmetries. Would you say that they are not fundamental things in themselves, or 'building-blocks of the universe,' to use the old-fashioned language?

Again, the difficulty is in the meaning of the words. Words like building blocks or really existing are too indefinite in their meaning, so I would hesitate to answer your questions, since an answer would depend on the definitions of the words.

DP To be more precise, ultimately could one have a description of nature which needed only elementary particles or, alternatively, a description in which the elementary particles would be defined in terms of the rest of the universe? Or is there no starting-point, as it were, no single axiom on which one can build the whole of physics?

No. Even if, for instance, that formula which Pauli and I wrote down fifty years ago turned out to be the correct formulation for the spectrum of elementary particles, it is certainly not the basis for all of physics. Physics can never be closed, or brought to an end, so that we must turn to biology or such things. What we can hope for, I think, is that we may get an explanation of the spectrum of elementary particles, and with it also an explanation of electromagnetism and gravitation, in the same sense as we get an explanation of the spectrum of a big molecule from the Schrödinger equation.

This does not mean that thereby physics has come to an end. It means that, for instance, at the boundary between physics and biology, there may be new features coming in which are not thought of in physics and chemistry. Something entirely new must happen when I try to use quantum theory within the realm of biology. Therefore I criticize those formulations which imply an end to physics.

DP Is it possible to reduce physics or any element of physics purely to logic and axioms?

I would say that certain parts of physics can always be reduced to logical mathematics or mathematical schemes. This has been possible for Newtonian physics, for quantum mechanics, and so on, so I do not doubt that it will also be possible for the world of the elementary particles. In astrophysics today, one comes upon pulsars and black holes, two regions in which gravitation becomes enormous, and perhaps a stronger force than all other forces. I could well imagine that in such black holes, for instance (if they exist), the spectrum of elementary particles would be quite different from the spectrum we now have. In the black holes, then, we would have a new area of physics, to some extent separated from that part which we now call elementary particle physics. There would be connections, and one would have to study how to go from the one to the other; but I do not believe in an end of physics.



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