Like any educated person of his time, Bohr knew of Kant's phenomemal/noumenal dualism. He often spoke as if the goal of complementarity was to reconcile opposites. He likened it to the eastern yin and yang, and his grave is marked with the yin/yang symbol.

Bohr was often criticized for suggesting that both A and Not-A could be the case. This was the characteristic sign of Georg W.F. Hegel's dialectical materialism. Had Bohr absorbed some Hegelian thinking? Another Hegelian trait was to speak indirectly and obscurely of the most important matters, and this was Bohr's way, to the chagrin of many of his disciples. They called it "obscure clarity." They hoped for clarity and but got mostly fuzzy thinking when Bohr stepped outside of his "old" quantum mechanics.

Bohr might very much have liked the current two-stage model for free will incorporating both randomness and an adequate statistical determinism. He could have seen it as a shining example of his complementarity.

As a philosopher, Bohr was a logical positivist, greatly influenced by Ernst Mach. He put severe epistemological limits on knowing the Kantian "things in themselves," just as Immanuel Kant had put limits on reason. The British empiricist philosophers John Locke and David Hume had put the "primary" objects beyond the reach of our "secondary" sensory perceptions. In this respect, Bohr shared the positivist views of many other empirical scientists, especially Mach.

Bohr seemed to deny the existence of an "objective reality," but clearly knew and said that the physical world is largely independent of human observations. In classical physics, the physical world is assumed to be completely independent of the act of observing the world.

Copenhageners were proud of their limited ability to know. Bohr said:

There is no quantum world. There is only an abstract quantum physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.

Quoted by Aage Petersen, Bulletin of the Atomic Scientists. Sep 1963, Vol. 19 Issue 7, p.12

Agreeing with many twentieth-century analytic language philosophers, Bohr and Heisenberg emphasized the importance of conventional language as a tool for knowledge. Since language evolved to describe the familiar world of "classical" objects in space and time, they insisted that somewhere between the quantum world and the classical world there must come a point when our observations and measurements can be expressible in classical concepts. They argued that a measurement apparatus and a particular observation must be describable classically in order for it to be understood and become knowledge in the mind of the observer. And controversially, they maintained that a measurement is not complete until it is knowledge in the mind of a "conscious observer."

In quantum physics, Bohr and Heisenberg said that the result of an experiment depends on the free choice of the experimenter as to what to measure. The quantum world of photons and electrons might look like waves or look like particles depending on what we look for, rather than what they "are" as "things in themselves."

The Copenhagen Interpretation of quantum mechanics says that physical reality does not actually have values for familiar physical variables like position and momentum until we decide to measure them. Einstein, by comparison, asked "Does the particle have a precise position the moment before it is measured?" The Copenhagen answer was sometimes "no," more often it was "we don't know," or "the question is meaningless."

Einstein, with his Princeton colleagues Boris Podolsky and Nathan Rosen, are said to have claimed in their article on what is now called the EPR paradox, that quantum mechanics requires the addition of further parameters or "hidden variables" to restore a deterministic picture of the "elements of reality."

In classical physics, such elements of reality include simultaneous values for the position and momentum of elementary particles like electrons.

In quantum mechanics, Bohr and Werner Heisenberg claimed that such properties could not be said to exist precisely before an experimenter decides to make a measurement.

This "freedom of choice" of the experimenter includes the freedom of which specific property to measure for. If the position is measured accurately, the (complementary conjugate and non-commuting variable) momentum is necessarily (sic) indeterminate.

For example, if the observer orients a Stern-Gerlach device in a particular direction or angle, that choice of direction limits the outcomes of measurement to spin-up or spin-down in that chosen direction. But the observer cannot control the random outcomes of spin-up or spin-down. This randomness is what Paul Dirac called "Nature's choice."

In his 1938 essay on "The Causality Problem in Atomic Physics" Bohr again emphasizes the "free choice" of an experimental procedure in his solution to the EPR paradox.

the paradox finds its complete solution within the frame of the quantum mechanical formalism, according to which no well defined use of the concept of "state" can be made as referring to the object separate from the body with which it has been in contact, until the external conditions involved in the definition of this concept are unambiguously fixed by a further suitable control of the auxiliary body. Instead of disclosing any incompleteness of the formalism, the argument outlined entails in fact an unambiguous prescription as to how this formalism is rationally applied under all conceivable manipulations of the measuring instruments. The complete freedom of the procedure in experiments common to all investigations of physical phenomena, is in itself of course contained in our free choice of the experimental arrangement, which again is only dictated by the particular kind of phenomena we wish to investigate.

(in "Causality and Complementarity," vol. IV of The Philosophical Writings of Niels Bohr, p. 102)

To be sure, those quantum events that are "measured" in a physics experiment which is set up to measure a certain quantity are dependent on the experimenter and the design of the experiment. To measure the electron spin in a Stern-Gerlach experiment, for example, the experimenter is "free to choose" to measure, for example, the z-component of the spin, rather than the x- or y-component. This will influence (but not determine) quantum level events in the following ways:

• The experimental outcome will produce a definite value for the z-component of the spin (either +1/2 or -1/2)

• The x-component of the spin after the measurement will be in a linear combination/superposition of +1/2 or -1/2 states
  | ψ > = (1/√2) | +1/2 > + (1/√2) | -1/2 >

On the other hand, we could not create the particular value for the position. This is a random choice made by Nature, as P. A. M. Dirac put it.

For many years, Bohr described these freedoms as a consequnce of "uncertainty," as in Heisenberg's famous "uncertainty principle." Bohr initially described this as an epistemological problem. Heisenberg's first explanation assumed that the measuring apparatus "disturbed" a particle in the act of measurement so that indeterminacy was ontological and independent of the observer.

The popular but mistaken thought experiment known as "Heisenberg's Microscope" showed that low-energy long-wavelength photons would not disturb an electron's momentum, but their long waves provided a blurry picture at best, so they lacked the resolving power to measure the position accurately. Conversely, if a high-energy, short wavelength photon was used (e.g., a gamma-ray), it might measure momentum accurately, but the recoil of the electron would be so large that its position becomes uncertain.

But Bohr showed Heisenberg was mistaken. One could correct for the disturbance (the recoil), he said, but could not ever eliminate the limits on resolving power of the measuring instrument, which is the real source of uncertainty. In his later years, Bohr stopped describing Heisenberg's principle as "uncertainty" and referred to it as "indeterminacy," the word Heisenberg himself had originally used (unbestimmtheit).

Δt is the time it takes the wave packet to pass a certain point.
Δν is the range of frequencies of the superposed waves.
In space instead of time, the wave packet is length Δx
and the range of waves per centimeter is Δσ

Bohr showed that a "wave-packet" with significant values in a spatially limited volume can be made from a superposition of plane waves with a range of frequencies.

The range of frequencies Δν needed so the wave packet is kept inside an interval of time Δt is

A similar argument in space relates the physical size of a wave packet Δx to the variation in the number of waves per centimeter Δσ. σ is the so-called wave number = 1 / λ (the inverse of the wavelength):

If we multiply both sides of the above equations by Planck's constant h, and use the Planck/Einstein relation between energy and frequency E = hν (and the similar Einstein relation between momentum and wavelength p = hσ = h / λ), the above become the Heisenberg indeterminacy relations:

This must have dazzled and perhaps upset Heisenberg. Bohr had used only the abstract space and time properties of waves to derive Heisenberg's physical limits! Bohr was obviously impressed by the new de Broglie - Schrödinger wave mechanics. Could wave mechanics produce a theory that did not need Heisenberg's material interactions or Einstein's point-like particles of energy (now photons)?

Bohr was pleased that Schrödinger's wave function provided a "natural" explanation for the "quantum numbers" of the "stationary states" in his first quantum postulate. They are simply the nodes in the wave function. On the other hand, Schrödinger hoped to eliminate the "unnatural" quantum jumps in Bohr's second quantum postulate by resonances or singularities in the wave field.

Quantum mechanics requires a fundamental "indeterminacy" that is ontological, a characteristic of the wave function whether or not it is observed. The experimenter can get different results, depending on the choice of measurement apparatus and the property or attribute measured.

EPR argued (mistakenly) that entangled particles could be regarded as separate systems (the indistinguishable particles are in fact described by an inseparable two-particle wave function), and since the experimenter can choose which type of measurement to make on the first system, it would make an instantaneous difference in the state and properties of the second system, however far away, without in any way "disturbing" the second system, but violating special relativity.

We see therefore that, as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems. Thus, it is possible to assign two different wave functions to the same reality (the second system after the interaction with the first).

(Physical Review, 47, 777, 1935))

In his 1935 reply to Einstein, Podolsky, and Rosen, Bohr denied that the limitations on simultaneously measuring complementary properties implied any incompleteness:

My main purpose in repeating these simple, and in substance well-known considerations, is to emphasize that in the phenomena concerned we are not dealing with an incomplete description characterized by the arbitrary picking out of different elements of physical reality at the cost of sacrificing other such elements, but with a rational discrimination between essentially different experimental arrangements and procedures which are suited either for an unambiguous use of the idea of space location or for a legitimate application of the conservation theorem of momentum.

Here Bohr introduces the freedom of the experimenter

Any remaining appearance of arbitrariness concerns merely our freedom of handling the measuring instruments characteristic of the very idea of experiment. In fact, the renunciation in each experimental arrangement of the one or the other of two aspects of the description of physical phenomena, - the combination of which characterizes the method of classical physics, and which therefore in this sense may be considered as complementary to one another, - depends essentially on the impossibility in the field of quantum theory, of accurately controlling the reaction of the object on the measuring instruments, i.e., the transfer of momentum in case of position measurements, and the displacement in case of momentum measurements.

Not only epistemological human ignorance of values, but even definitions of physical quantities are an impossibility.

Just in this last respect any comparison between quantum mechanics and ordinary statistical mechanics, - however useful it may be for the formal presentation of the theory, — is essentially irrelevant. Indeed we have in each experimental arrangement suited for the study of proper quantum phenomena not merely to do with an ignorance of the value of certain physical quantities, but with the impossibility of defining these quantities in an unambiguous way.

The last remarks apply equally well to the special problem treated by Einstein, Podolsky and Rosen, which has been referred to above, and which does not actually involve any greater intricacies than the simple examples discussed above. The particular quantum-mechanical state of two free particles, for which they give an explicit mathematical expression, may be reproduced, at least in principle, by a simple experimental arrangement, comprising a rigid diaphragm with two parallel slits, which are very narrow compared with their separation, and through each of which one particle with given initial momentum passes independently of the other. If the momentum of this diaphragm is measured accurately before as well as after the passing of the particles, we shall in fact know the sum of the components perpendicular to the slits of the momenta of the two escaping particles, as well as the difference of their initial positional coordinates in the same direction; while of course the conjugate quantities, i.e., the difference of the components of their momenta, and the sum of their positional coordinates, are entirely unknown.* In this arrangement, it is therefore clear that a subsequent single measurement either of the position or of the momentum of one of the particles will automatically determine the position or momentum, respectively, of the other particle with any accuracy; at least if the wave-length corresponding to the free motion of each particle is sufficiently short compared with the width of the slits. As pointed out by the named authors, we are therefore faced at this stage with a completely free choice whether we want to determine the one or the other of the latter quantities by a process which does not directly interfere with the particle concerned.

Like the above simple case of the choice between the experimental procedures suited for the prediction of the position or the momentum of a single particle which has passed through a slit in a diaphragm, we are, in the "freedom of choice" offered by the last arrangement, just concerned with a discrimination between different experimental procedures which allow of the unambiguous use of complementary classical concepts. In fact to measure the position of one of the particles can mean nothing else than to establish a correlation between its behavior and some instrument rigidly fixed to the support which defines the space frame of reference. Under the experimental conditions described such a measurement will therefore also provide us with the knowledge of the location, otherwise completely unknown, of the diaphragm with respect to this space frame when the particles passed through the slits. Indeed, only in this way we obtain a basis for conclusions about the initial position of the other particle relative to the rest of the apparatus. By allowing an essentially uncontrollable momentum to pass from the first particle into the mentioned support, however, we have by this procedure cut ourselves off from any future possibility of applying the law of conservation of momentum to the system consisting of the diaphragm and the two particles and therefore have lost our only basis for an unambiguous application of the idea of momentum in predictions regarding the behavior of the second particle. Conversely, if we choose to measure the momentum of one of the particles, we lose through the uncontrollable displacement inevitable in such a measurement any possibility of deducing from the behavior of this particle the position of the diaphragm relative to the rest of the apparatus, and have thus no basis whatever for predictions regarding the location of the other particle.

From our point of view we now see that the wording of the above-mentioned criterion of physical reality proposed by Einstein, Podolsky and Rosen contains an ambiguity as regards the meaning of the expression "without in any way disturbing a system."

EPR was concerned about faster-than-light disturbances or influences between particles with pre-existing properties

Of course there is in a case like that just considered no question of a mechanical disturbance of the system under investigation during the last critical stage of the measuring procedure. But even at this stage there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system. Since these conditions constitute an inherent element of the description of any phenomenon to which the term "physical reality" can be properly attached,

Bohr denies the charge of "incompleteness"

we see that the argumentation of the mentioned authors does not justify their conclusion that quantum-mechanical description is essentially incomplete. On the contrary this description, as appears from the preceding discussion, may be characterized as a rational utilization of all possibilities, of unambiguous interpretation of measurements, compatible with the finite and uncontrollable interaction between the object and the measuring instruments in the field of quantum theory. In fact, it is only the mutual exclusion of any two experimental procedures, permitting the unambiguous definition of complementary physical quantities, which provides room for new physical laws the coexistence of which might at first sight appear irreconcilable with the basic principles of science. It is just this entirely new situation as regards the description of physical phenomena that the notion of complementarity aims at characterizing.

(Physical Review, 48, 696, 1935))

the paradox finds its complete solution within the frame of the quantum mechanical formalism, according to which no well defined use of the concept of "state" can be made as referring to the object separate from the body with which it has been in contact, until the external conditions involved in the definition of this concept are unambiguously fixed by a further suitable control of the auxiliary body. Instead of disclosing any incompleteness of the formalism, the argument outlined entails in fact an unambiguous prescription as to how this formalism is rationally applied under all conceivable manipulations of the measuring instruments. The complete freedom of the procedure in experiments common to all investigations of physical phenomena, is in itself of course contained in our free choice of the experimental arrangement, which again is only dictated by the particular kind of phenomena we wish to investigate.

(in "Causality and Complementarity," vol. IV of The Philosophical Writings of Niels Bohr, p. 102)

In all the recent EPR experiments to test Bell's Inequalities, "free choices" of the experimenters are needed when they select the angle of polarization. Note that what determines the second experimenter's results is these tests is simply the first experimenter's measurement, which instantaneously collapses the superposition of two-particle states into a particular state that is now a separable product of independent particle states.

Bell inequality investigators who try to recover the "elements of local reality" that Einstein wanted, and who hope to eliminate the irreducible randomness of quantum mechanics that follows from wave functions as probability amplitudes, often cite "loopholes" in EPR experiments. For example, the "detection loophole" claims that the efficiency of detectors is so low that they are missing many events that might prove Einstein was right.

Most all the loopholes have now been closed, but there is one loophole that can never be closed because of its metaphysical/philosophical nature. That is the "(pre-)determinism loophole."

If every event occurs for reasons that were established at the beginning of the universe, then the experimenters lack any free will or free choice and all the careful experimental results are meaningless. John Conway and Simon Kochen have formalized this loophole in what they call the Free Will Theorem.