Objective Reality and Entanglement

Here is my four-year-old animation of the entanglement experiment …

Now consider this possible animation of the experiment, which illustrates the assumption that each electron is in a linear combination of up and down spin. It imitates the superposition (or linear combination) with up and down arrows on each electron oscillating quickly.

Notice that if you mouse click on any frame in the timeline, you will see that total spin = 0 is conserved. When one electron is spin up the other is always spin down. But what justifies the assumption that the spin of each electron is in a superposition of up and down? For the Copenhagen Interpretation, it is simply that we do not know which it is.

Since quantum mechanics says we cannot know the spin until it is measured, our best estimate is a 50/50 probability between up and down.

This is the same as assuming Schrödinger’s Cat is 50/50 alive and dead. But what this means of course is simply that if we do a large number of identical experiments, the statistics for live and dead cats will be approximately 50/50%. We never observe/measure a cat that is both dead and alive!

As Einstein noted, QM tells us nothing about individual cats. Quantum mechanics is incomplete in this respect. He is correct, although Bohr and Heisenberg insisted QM is complete, because we cannot know more before we measure, and reality is created (they say) when we do measure.

Despite accepting that a particular value of an “observable” can only be known by a measurement (knowledge is an epistemological problem, Einstein asked whether the particle actually (really, ontologically) has a path and position before we measure it? His answer was yes.

Einstein believed strongly in conservation principles – conservation of energy, momentum, angular momentum, and spin. Once a particle is on a path, with a spin, what could possibly change its spin mid-flight to its detection?

Here is an animation that assumes the two electrons are randomly produced in a spin-up and a spin-down state. Einstein’s objective reality suggest that they remain in those states no matter how far they separate, provided neither interacts (is decohered) until the measurement. Too simple to be true?

Almost every presentation of the EPR paradox begins with something like “Alice observes one particle…” and concludes with the question “How does the second particle get the information needed so that Bob’s measurements correlate perfectly with Alice?”

There is a fundamental asymmetry in this framing of the EPR experiment. It is a surprise that Einstein, who was so good at seeing deep symmetries, did not consider how to remove the asymmetry.

Consider this reframing: Alice’s measurement collapses the two-particle wave function. The two indistinguishable particles simultaneously appear at locations in a space-like separation. The frame of reference in which the source of the two entangled particles and the two experimenters are at rest is a special frame in the following sense.

As Einstein knew very well, there are frames of reference moving with respect to the laboratory frame of the two observers in which the time order of the events can be reversed. In some moving frames Alice measures first, but in others Bob measures first.

If there is a special frame of reference (not a preferred frame in the relativistic sense), surely it is the one in which the origin of the two entangled particles is at rest. Assuming that Alice and Bob are also at rest in this special frame and equidistant from the origin, we arrive at the simple picture in which any measurement that causes the two-particle wave function to collapse makes both particles appear simultaneously at determinate places with fully correlated properties (just those that are needed to conserve energy, momentum, angular momentum, and spin).

Individuation and the Necessity of Identity

Since at least the time of Aristotle, philosophers have debated what it is that constitutes an individual person or thing. What makes it a unity, numerically one? What distinguishes it from everything else?

Individuation is related to the metaphysical problems of constitution, composition, colocation, essentialism, and identity.

Given two equal amounts of matter, they are distinguished by their shape or form. Given two things with identical form, they are individuated by being embodied in different material.

In information philosophy, identity depends on the total information in an object or concept.

We distinguish the intrinsic information inside the object (or concept) from any relational information with respect to other objects that we call extrinsic or external information. We can “pick out” the intrinsic information as that which is “self-identical” in an object. The Greeks called this the πρὸς ἑαυτο – self-relation. or ἰδίος ποιὸν, “peculiar qualifications” of the individual.

Self-identity, then, is the simple fact that the intrinsic information and the extrinsic relational or dispositional information are unique to this single object. No other object can have the same disposition relative to other objects. This is an absolute kind of identity. Some metaphysicians say that such identity is logically necessary. Some say self-identity is the only identity, but we can now support philosophers who argue for a relative identity.

To visualize our concept of information identity, imagine putting yourself in the position of an object. Look out at the world from its vantage point. No other object has that same view, that same relation with the objects around you, especially its relation with you. Now another object could have intrinsic information identicality. We will identify a very large number of objects and concepts in the world that are intrinsically identical, including natural and artifactual kinds, which we may call digital kinds, since they are identical, bit for bit. This is relative identity.

In 1947, Ruth C. Barcan (later Ruth Barcan Marcus) wrote an article on “The Identity of Individuals.” It was the first assertion of the so-called “necessity of identity.” Her work was written in the dense expressions of symbolic logic, with little explanation.

Five years later, Marcus’s thesis adviser, Frederic B. Fitch, published his book, Symbolic Logic, which contained the simplest proof ever of the necessity of identity, by the simple mathematical substitution of b for a in the necessity of self-identity statement (2).

23.4

(1) a = b,

(2) ☐[a = a], then

(3) ☐[a = b], by identity elimination. 

René Descartes: The Origins of the Mind-Body Problem

In his 1644 Principles of Philosophy, Descartes identified freedom with actions that are not pre-determined, even by the existence of divine foreknowledge.

Descartes was the origin of the Mind-Body Problem. He famously divided the world into mind (the ideal realm of thoughts) and body (the material world). For him, the physical world was a deterministic machine, but our ideas and thoughts could be free (undetermined) and could change things in the material world (through the pineal gland in the brain, he thought).

Here are the relevant sections in Descartes’ Principles.

37. The supreme perfection of man is that he acts freely or voluntarily, and it is this which makes him deserve praise or blame.The extremely broad scope of the will is part of its very nature. And it is a supreme perfection in man that he acts voluntarily, that is, freely; this makes him in a special way the author of his actions and deserving of praise for what he does. We do not praise automatons for accurately producing all the movements they were designed to perform, because the production of these movements occurs necessarily. It is the designer who is praised for constructing such carefully-made devices; for in constructing them he acted not out of necessity but freely. By the same principle, when we embrace the truth, our doing so voluntarily is much more to our credit than would be the case if we could not do otherwise.

39. The freedom of the will is self-evident.

There is freedom in our will, and that we have power in many cases or withhold our assent at will, is so evident that it must be counted among the first and most common notions that are innate in us. This was obvious earlier on when, in our attempt to doubt everything, we went so far as to make the supposition of some supremely powerful author of our being who was attempting to deceive us in every possible way. For in spite of that supposition, the freedom which we experienced within us was nonetheless so great as to enable us to abstain from believing whatever was not quite certain or fully examined. And what we saw to be beyond doubt even during the period of that supposition is as self-evident and as transparently clear as anything can be.

40. It is also certain that everything was preordained by God.

But now that we have come to know God, we perceive in him a power so immeasurable that we regard it as impious to suppose that we could ever do anything which was not already preordained by him. And we can easily get ourselves into great difficulties if we attempt to reconcile this divine preordination with the freedom of our will, or attempt to grasp both these things at once.

41. How to reconcile the freedom of our will with divine preordination.But we shall get out of these difficulties if we remember that our mind is finite, while the power of God is infinite — the power by which he not only knew from eternity whatever is or can be, but also willed it and preordained it. We may attain sufficient knowledge of this power to perceive clearly and distinctly that God possesses it; but we cannot get sufficient grasp of it to see how it leaves the free actions of men undetermined. Nonetheless, we have such close awareness of the freedom and indifference1 which is in us, that there is nothing we can grasp evidently or more perfectly. And it would be absurd, simply because we do not grasp one thing, which we know must by its very nature be beyond our comprehension, to doubt something else of which we have intimate grasp and which we experience within ourselves.

42. Although we do not want to go wrong, nevertheless we go wrong by our own will.

Now that we know that all our errors depend on the will, it may surprising that we should ever go wrong, since there is no one who wants to go wrong. But there is a great difference between choosing to go wrong and choosing to give one’s assent in matters where, as it happens, error is to be found. And although there is in fact no one who expressly wishes to go wrong, there is scarcely anyone who does not often wish to assent to something which, though he does not know it, contains some error. Indeed, precisely because of their eagerness to find the truth, people who do not know the right method of finding it often pass judgement on things of which they lack perception, and this is why they fall into error.

If Descartes did equate freedom and indeterminacy with the mind, and determinism with the body, it would anticipate and be consistent with Kant‘s later view that the realm of freedom is noumenal, where phenomena are determined, for Kant by Newton’s deterministic laws of nature.

Chance in Philosophy, Physics, Biology, Psychology, and Neuroscience

Chance is often defined as the opposite of Necessity. The English word derives from the Latin cadere – to fall, especially cadens a fall, falling. Dictionary definitions refer to the fall of the dice, but the etymology suggests it is related to the grammatical idea of declension, which describes the falling or “leaning” away of the genitive, dative, and accusative cases from the “straight up” nominative case. The word connotes falling in the sense of decadence. Note the German for chance is Zufall.

 

Leucippus (440 B.C.E.) stated the first dogma of determinism, an absolute necessity.

Nothing occurs by chance (maton), but there is a reason (logos) and necessity (ananke) for everything.

Chance is regarded as inconsistent with causal determinism and with physical or mechanicaldeterminism.

The idea that Chance and Necessity are the only two logical options, and that neither is compatible with free will and moral responsibility, is the basis for the standard argument against free will.

The first thinker to suggest a physical explanation for chance in the universe was Epicurus. Epicurus was influenced strongly by Aristotle, who regarded chance as a fifth cause. He said there must be cases in which the normally straight paths of atoms in the universe occasionally bend a little and the atoms “swerve” to prevent the universe and ourselves from being completely determined by the mechanical laws of Democritus.

For Epicurus, the chance in his atomic swerve was simply a means to deny the fatalistic future implied by determinism (and necessity). As the Epicurean Roman Lucretius explained the idea,

…if all motion is always one long chain, and new motion arises out of the old in order invariable, and if the first-beginnings do not make by swerving a beginning of motion such as to break the decrees of fate, that cause may not follow cause from infinity, whence comes this freedom in living creatures all over the earth
(De Rerum Natura, Book 2, lines 251-256)

Epicurus did not say the swerve was directly involved in decisions so as to make them random. His critics, ancient and modern, have claimed mistakenly that Epicurus did assume “one swerve – one decision.” Some recent philosophers call this the “traditional interpretation” of Epicurean free will. On the contrary, following Aristotle, Epicurus thought human agents have an autonomous ability to transcend the necessity and chance of some events. This special ability makes us morally responsible for our actions.

Epicurus, clearly following Aristotle, finds a tertium quid, beyond necessity (Democritus’ physics) and chance (Epicurus’ swerve). 
The tertium quid is agent autonomy

…some things happen of necessity (ἀνάγκη), others by chance (τύχη), others through our own agency (παρ’ ἡμᾶς).
…necessity destroys responsibility and chance is uncertain; whereas our own actions are autonomous, and it is to them that praise and blame naturally attach.

λέγει ἐν ἄλλοις γίνεσθαι ἃ μὲν κατ’ ἀνάγκην, ἃ δὲ ἀπὸ τύχης, ἃ δὲ παρ’ ἡμᾶς, διὰ τὸ τὴν μὲν ἀνάγκην ἀνυπεύθυνον εἶναι, τὴν δὲ τύχην ἄστατον ὁρᾶν, τὸ δὲ παρ’ ἡμᾶς ἀδέσποτον, ᾧ καὶ τὸ μεμπτὸν καὶ τὸ ἐναντίον παρακολουθεῖν πέφυκεν 
(Letter to Menoeceus, §133)

Despite abundant evidence, many philosophers deny that real chance exists. If a single event is determined by chance, then indeterminism would be true, they say, and undermine the very possibility of certain knowledge. Some go to the extreme of saying that chance makes the state of the world totally independent of any earlier states, which is nonsense, but it shows how anxious they are about chance.

The Problem of the Soul, Self, Spirit, Mind, and “Ghost in the Machine”

Celebrating René Descartes, the first modern philosopher, and his famous phrase Ego cogito, ergo sum, we call our model for mind the Ego. It is implemented with our experience recorder and reproducer (ERR).

Our two-stage model for free will we call the Cogito. Our model for an objective value, independent of humanity and earthly bioethics, we call Ergo. And our model for knowledge we call the Sum.

The Ego is more or less synonymous with the Soul, the Self, or the Spirit – Gilbert Ryle’s “ghost in the machine.” We see it as immaterial information. An immaterial self with causal power is almost universally denied by modern philosophers as metaphysical, along with related problematic ideas such as consciousness and libertarian or indeterministic free will.

Descartes’ suggestion that animals are machines included the notion that man too is in part a machine – the human body obeys deterministic causal laws. Although for Descartes man also has a soul or spirit or mind that is exempt from determinism and thus from what is known today as “causal closure,” Cartesian dualism was the first step to eliminative materialism. 

But as all critics of Descartes do, we must ask, how can the mind both cause something physical to happen and yet itself be acausal,? How is it exempt from causal chains coming up from the body?

Descartes’ vision of undetermined freedom for the mind is realized since our immaterial thoughts are free, whereas our actions are adequately determined by our will. This combination of ideas is the basis for our two-stage model of free will.1 It is a model of agent causation. New causal chains originate as ideas in our minds. Once evaluated and chosen they are adequately determined to lead to willed actions. This is a model for self-determination.

The “self ” or ego, the psyche or soul, is the self of this self-determination. Self-determination is of course limited by our control over matter and energy, but within those physical constraints our selves can consider ideas, decide to act on one and take full responsibility for our actions.

The Self is often identified with one’s “character.” This is the basis for saying that our choices and decisions are made by evaluating freely generated alternative possibilities in accordance with our reasons, motives, feelings, desires, etc. These are in turn often the consequence of our past experiences, along with inherited (biologically built-in) preferences. And this bundle of motivating factors is essentially what is known as our character. Someone familiar with all of those preferences would be able to predict our actions with some certainty, though not perfectly, when faced with particular options and the circumstances. The self is the agent that is the cause for those actions.

 

Einstein to Heisenberg: What, No Paths? What, No Photons?

Heisenberg tells us that in 1926, Einstein asked him about Einstein’s theory of light quanta (photons). At that time, Einstein’s radical theory was 21 years old, and had been accepted by almost all physicists, because it had explained the Compton effect in 1923 and had disproved the Bohr-Kramers-Slater theory, which denied photons, of 1924.

[Heisenberg described their talk.] On the way, he asked about my studies and previous research. As soon as we were indoors, he opened the conversation with a question that bore on the philosophical background of my recent work.

“What you have told us sounds extremely strange. You assume the existence of electrons inside the atom, and you are probably quite right to do so. But you refuse to consider their orbits, even though we can observe electron tracks in a cloud chamber. I should very much like to hear more about your reasons for making such strange assumptions.”

Heisenberg explains that he substituted the observable frequencies of spectral line emissions – as “representatives” of the unobservable electron orbits. But there is a great difference between not being able to observe electron paths and declaring they do nor exist.

“We cannot observe electron orbits inside the atom,” I must have replied, “but the radiation which an atom emits during discharges enables us to deduce the frequencies and corresponding amplitudes of its electrons. After all, even in the older physics wave numbers and amplitudes could be considered substitutes for electron orbits. Now, since a good theory must be based on directly observable magnitudes, I thought it more fitting to restrict myself to these, treating them, as it were, as representatives of the electron orbits…”

“But what happens during the emission of light? As you know, I suggested that, when an atom drops suddenly from one stationary energy value to the next, it emits the energy difference as an energy packet, a so-called light quantum. In that case, we have a particularly clear example of discontinuity. Do you think that my conception is correct? Or can you describe the transition from one stationary state to another in a more precise way?”

In my reply, I must have said something like this:

[Heisenberg says simply that he and Bohr “Do not know.” He cannot say that he believes in Einstein’s light quanta, although by this time most quantum physicists had come to accept the ides of photons as particles, as well as their having wave properties!]

“Bohr has taught me that one cannot describe this process by means of the traditional concepts, i.e., as a process in time and space. With that, of course, we have said very little, no more, in fact, than that we do not know. Whether or not I should believe in light quanta, I cannot say at this stage. Radiation quite obviously involves the discontinuous elements to which you refer as light quanta.

[Heisenberg could not then see how his quantum mechanics, with its emphasis on the material particle properties of energy and momentum, can explain wave properties, which Bohr sees as described in terms of the complementaryproperties of space and time.]

“On the other hand, there is a continuous element, which appears, for instance, in interference phenomena, and which is much more simply described by the wave theory of light. But you are of course quite right to ask whether quantum mechanics has anything new to say on these terribly difficult problems. I believe that we may at least hope that it will one day.

“I could, for instance, imagine that we should obtain an interesting answer if we considered the energy fluctuations of an atom during reactions with other atoms or with the radiation field. If the energy should change discontinuously, as we expect from your theory of light quanta, then the fluctuation, or, in more precise mathematical terms, the mean square fluctuation, would be greater than if the energy changed continuously. I am inclined to believe that quantum mechanics would lead to the greater value, and so establish the discontinuity. On the other hand, the continuous element, which appears in interference experiments, must also be taken into account. Perhaps one must imagine the transitions from one stationary state to the next as so many fade-outs in a film. The change is not sudden—one picture gradually fades while the next comes into focus so that, for a time, both pictures become confused and one does not know which is which. Similarly, there may well be an intermediate state in which we cannot tell whether an atom is in the upper or the lower state.”

[Einstein is quite correct that Heisenberg is talking about what we subjectively know—epistemology— and not about what is—ontology—what is going on in objective reality.]

“You are moving on very thin ice,” Einstein warned me. “For you are suddenly speaking of what we know about nature and no longer about what nature really does. In science we ought to be concerned solely with what nature does. It might very well be that you and I know quite different things about nature. But who would be interested in that? Perhaps you and I alone. To everyone else it is a matter of complete indifference. In other words, if your theory is right, you will have to tell me sooner or later what the atom does when it passes from one stationary state to the next.”

REFERENCES

Albert Einstein

Werner Heisenberg

Entanglement

The Mind and the Quantum Wave Function Are Both Pure Abstract Information. Is This Panpsychism?

Light waves are often compared to water waves. Quantum probability waves, but this latter is a serious error. Water waves and light waves (as well as sound waves) contain something substantial like matter or energy. But quantum waves are just abstract information – mathematical possibilities.

The quantum probability amplitude is pure information. It is neither matter nor energy. When a wave function “collapses” or “goes through both slits” in the dazzling two-slit experiment, nothing material is traveling faster than the speed of light or going through both slits.

Open with motion control

We shall argue that the particle of matter or energy always goes through just one slit, although the popular Copenhagen interpretation of physics claims we cannot know the particle path, that a path does not even exist until we make a measurement, that the particle may be in more than one place at the same time, and other similar nonsense that deeply bothered Einstein as he hoped for an “objective reality.”

A large number of panpsychists, some philosophers, and some scientists, believe that the mind of a conscious observer is needed to cause the collapse of the wave function.

Open with motion control

The Conscious Observer in Quantum Mechanics

Alfred North Whitehead: Continuous Infinite Fields or Finite Discrete Particles?

Alfred North Whitehead was an English mathematician (best known among scientists for his work with his student Bertrand Russell on the Principia Mathematica). But in philosophy and theology, Whitehead is best known as a philosopher whose later work at Harvard included his Process Philosophy and the subsequent development of a Process Theology.

At Harvard, Whitehead supervised Willard van Orman Quine‘s Ph.D. thesis on the Russell and Whitehead Principia Mathematica. Russell and Quine would become giants in the twentieth-century fields of logical positivism and logical empiricism. Although logical positivism and later analytic language philosophy overwhelmed Whiteheadian “process” thinking in philosophy departments, Whitehead’s “process theology” has grown strong in divinity schools around the world. And “Whiteheadian” physicists. impressed by Whitehead’s analysis of events in space and time in special relativity as organic “occasions.” are prominent in debates about the role of quantum mechanics in consciousness and panpsychism.

Whitehead’s “philosophy of organism” analyzes the perception of experience as a continuing series of discrete “events” that are created and destroyed. He goes beyond the simple materialist view of elementary particles interacting in space and time, merely following the laws of classical and quantum mechanics. Beyond the atomic particles and the electromagnetic and gravitational fields, and beyond the conservation laws for energy and momentum, Whitehead sees an “organic” evolutionary process of creation and valuation.

We need to understand what it is exactly that Whitehead thinks is being created and why it can serve as a basis for values. We will argue that Whitehead’s process is “organic” because it explains evolution, not merely biological evolution but the cosmic evolution of the galaxies, stars, and planets as well as the creation of all matter from the primordial elementary particles.

In addition to his deep understanding of mathematics, Whitehead may have understood the development of modern physics better than any living philosopher in his day. He saw the greatest invention of the nineteenth century as the invention of the method of invention, namely the scientific method and newly created scientific information, but even more deeply, the means by which novel ideas of all kinds are created.

Whitehead identified four great novel ideas as the new nineteenth-century foundations of physical science, fields, particles, conservation principles, and evolution. The great question for Whitehead as a mathematician (and for Einstein as a physicist) was “Is nature continuous or discrete, fields or particles, infinities or a finite number of objects?”

Whitehead wrote in his great book Science and the Modern World,

One of the ideas is that of a field of physical activity pervading all space, even where there is an apparent vacuum. This notion had occurred to many people, under many forms. We remember the medieval axiom, nature abhors a vacuum…Thus in the seventies of the last century, some main physical sciences were established on a basis which presupposed the idea of continuity.

On the other hand, the idea of atomicity had been introduced by John Dalton, to complete Lavoisier’s work on the foundation of chemistry. This is the second great notion. Ordinary matter was conceived as atomic: electromagnetic effects were conceived as arising from a continuous field…The notion of matter as atomic has a long history. Democritus and Lucretius will at once occur to your minds. In speaking of these ideas as novel, I merely mean relatively novel,..In the eighteenth century every well-educated man read Lucretius, and entertained ideas about atoms. But John Dalton made them efficient in the stream of science; and in this function of efficiency atomicity was a new idea. The influence of atomicity was not limited to chemistry. The living cell is to biology what the electron and the proton are to physics.

The remaining pair of new ideas to be ascribed to this epoch are both of them connected with the notion of transition or change. They are the doctrine of the conservation of energy, and the doctrine of evolution.

The doctrine of energy has to do with the notion of quantitative permanence underlying change. The doctrine of evolution has to do with the emergence of novel organisms as the outcome of chance. The theory of energy lies in the province of physics. The theory of evolution lies mainly in the province of biology, although it had previously been touched upon by Kant and Laplace in connection with the formation of suns and planets.

Meaning and Information, Gottlob Frege’s Sense and Reference, Intension and Extension in Epistemology

In response to a commentor’s question, today’s lecture is more Great Problems and Metaphysics than Free Will. I am hoping that this lecture will also serve to show how the Information Philosopher website should work for its users.

Information philosophy hopes to go “beyond language and logic” because words are always ambiguous and dependent on their context.

Knowledge can be defined as information in minds that is a partial isomorphism (mapping) of the information structures in the external world. Information philosophy is a correspondence theory.

Sadly, there is no isomorphism, no information in common, between words and objects. This accounts for much of the failing of analytic language philosophy in the past century.

Although language is an excellent tool for human communication, it is arbitrary, ambiguous, and ill-suited to represent the world directly. Human languages do not picture reality. Information is the lingua franca of the universe.

The extraordinarily sophisticated connection between words and objects is made in human mindsmediated by the brain’s experience recorder and reproducer (ERR). Words stimulate neurons to start firing and to play back any similar experiences that include relevant objects and events.

Neurons that were wired together in our earliest experiences fire together at later times, contextualizing our new experiences, giving them meaning. And by replaying emotional reactions to those for similar earlier experiences, it makes then “subjective experiences,” giving us the feeling of “what it’s like to be me” and solving the “hard problem” of consciousness.

Gottlob Frege drew a distinction between the reference (denotation, name) and the sense (meaning, concept) of a word. But few know that Frege limited the “sense” to the everyday meaning attached to a word by the users of their language. Frege also described the “idea” or “representation” (Vorstellung) that would form in the mind of the message receiver. This, he said, would be different in every mind, since it is dependent on the peculiar experiences of each person. See Frege, Sense and Reference, p.212-213.

Frege’s distinction is the difference between intension and extension in the works of  many philosophers, including the difference between internalism and externalism in epistemology.

Getting to Know How to Use the Information Philosopher Website, Books, and Online Lectures

The goal of the Information Philosopher website is to provide free resources online that are beyond the reach of those without faculty academic privileges like mine.

It is also to teach a new method for the study of problems in philosophy, physics, biology, and psychology that goes beyond debating issues and interpretations. Words alone are too ambiguous to resolve the deepest questions facing us.

This new method of examining information structures, their communications and processing of information, provides new insight into the nature of reality and our place in the universe.

We all are dynamic and growing information processing structures. We can learn at every level in the human body, from the miniature molecular machines in our cells to the thinking in our minds/brains, how information controls the organization of matter one particle at a time.

Hundreds of webpages on thinkers and many more on problems are massively hyperlinked, allowing you to navigate effortlessly between closely related pages. It has a glossary of technical terms to help beginners appreciate the debating details.