Theory is Approximation

A farmer wants to increase milk production. They ask a physicist for advice. The physicist visits the farm, takes a lot of notes, draws some diagrams, then says, "OK, I need to do some calculations."

A week later, the physicist comes back and says, "I've solved the problem and I can tell you how to increase milk production".

"Great", says the farmer, "How?".

"First", says the physicist, "assume a spherical cow in a vacuum..."

What is Science?

Science is many things to many people. At times, scientists (or, at least, science enthusiasts) seem to claim that they alone know the truth of reality. Some seem to assume that "laws of science" are equivalent to laws of nature. Some go as far as stating that nature is governed by such "laws". 

Some believe that only scientific facts are true and that no metaphysics are possible. While this view is less common now, it was of major importance in the formulation of quantum theory, which still has problems admitting that reality exists. As Mara Beller (1996) notes:

Strong realistic and positivistic strands are present in the writings of the founders of the quantum revolution-Bohr, Heisenberg, Pauli and Born. Militant positivistic declarations are frequently followed by fervent denial of adherence to positivism (183). 

On the other hand, some see science as theory-laden and sociologically determined. Science is just one knowledge system amongst many of equal value. 

However, most of us understand that scientific theories are descriptive and idealised. And this is the starting point for me. 

In practising science, I had ample opportunity to witness hundreds or even thousands of objective (or observer-independent) facts about the world. The great virtue of the scientific experiment is that you get the same result, within an inherent margin of error associated with measurement, no matter who does the experiment or how many times they do it. The simplest explanation of this phenomenon is that the objective world exists and that such facts are consistent with reality. Thus, I take knowledge of such facts to constitute knowledge about reality. The usual label for this view is metaphysical realism.

However, I don't take this to be the end of the story. Realism has a major problem, identified by David Hume in the 1700s. The problem is that we cannot know reality directly; we can only know it through experience. Immanuel Kant's solution to this has been enormously influential. He argues that while reality exists, we cannot know it. In Kant's view, those qualities and quantities we take to be metaphysical—e.g. space, time, causality, etc.—actually come from our own minds. They are ideas that we impose on experience to make sense of it. This view is known as transcendental idealism. One can see how denying the possibility of metaphysics (positivism) might be seen as (one possible) extension of this view. 

It's important not to confuse this view with the idea that only mind is real. This is the basic idea of metaphysical idealism. Kant believed that there is a real world, but we can never know it. In my terms, there is no epistemic privilege.

Where Kant falls down is that he lacks any obvious mechanism to account for shared experiences and intersubjectivity (the common understanding that emerges from shared experiences). We do have shared experiences. Any scenario in which large numbers of people do coordinated movements can illustrate what I mean. For example, 10,000 spectators at a tennis match turning their heads in unison to watch a ball be batted back and forth. If the ball is not objective, or observer-independent, how do the observers manage to coordinate their movements? While Kant himself argues against solipsism, his philosophy doesn't seem to consider the possibility of comparing notes on experience, which places severe limits on his idea. I've written about this in Buddhism & The Limits of Transcendental Idealism (1 April 2016).

In a pragmatic view, then, science is not about finding absolute truths or transcendental laws. Science is about idealising problems in such a way as to make a useful approximation of reality. And constantly improving such approximations. Scientists use these approximations to suggest causal explanations for phenomena. And finally, we apply the understanding gained to our lives in the form of beliefs, practices, and technologies. 

What is an explanation?

In the 18th and 19th centuries, scientist confidently referred to their approximations as "laws". At the time, a mechanistic universe and transcendental laws seemed plausible. They were also gathering the low-hanging fruit, those processes which are most obviously consistent and amenable to mathematical treatment. By the 20th century, as mechanistic thinking waned, new approximations were referred to as "theories" (though legacy use of "law" continued). And more recently, under the influence of computers, the term "model" has become more prevalent. 

A scientific theory provides an explanation for some aspect of reality, which allows us to understand (and thus predict) how what we observe will change over time. However, even the notion of explanation requires some unpacking.

In my essay, Does Buddhism Provide Good Explanations? (3 Feb 23), I noted Faye's (2007) typology of explanation:

• Formal-Logical Mode of Explanation: A explains B if B can be inferred from A using deduction.
• Ontological Mode of Explanation: A explains B if A is the cause of B.
• Pragmatic Mode of Explanation: a good explanation is an utterance that addresses a particular question, asked by a particular person whose rational needs (especially for understanding) must be satisfied by the answer.

In this essay, I'm striving towards the pragmatic mode and trying to answer my own questions. 

Much earlier (18 Feb 2011), I outlined an argument by Thomas Lawson and Robert McCauley (1990) which distinguished explanation from interpretation.

• Explanationist: Knowledge is the discovery of causal laws, and interpretive efforts simply get in the way.
• Interpretationist: Inquiry about human life and thought occurs in irreducible frameworks of values and subjectivity. 

"When people seek better interpretations they attempt to employ the categories they have in better ways. By contrast, when people seek better explanations they go beyond the rearrangement of categories; they generate new theories which will, if successful, replace or even eliminate the conceptual scheme with which they presently operate." (Lawson & McCauley 1990: 29)

The two camps are often hostile to each other, though some intermediate positions exist between them. As I noted, Lawson and McCauley see this as somewhat performative:

Interpretation presupposes a body of explanation (of facts and laws), and seeks to (re)organise empirical knowledge. Explanation always contains an element of interpretation, but successful explanations winnow and increase knowledge. The two processes are not mutually exclusive, but interrelated, and both are necessary.

This is especially true for physics where explanations often take the form of mathematical equations that don't make sense without commentary/interpretation.  

Scientific explanation.

Science mainly operates, or aims to operate, in the ontological/causal mode of explanation: A explains B if (and only if) A is the cause of B. However, it still has to satisfy the conditions for being a good pragmatic explanation:  "a good explanation is an utterance that addresses a particular question, asked by a particular person whose rational needs (especially for understanding) must be satisfied by the answer."

As noted in my opening anecdote, scientific models are based on idealisation, in which an intractably complex problem is idealised until it becomes tractable. For example, in kinematic problems, we often assume that the centre of mass of an object is where all the mass is. It turns out that when we treat objects as point masses in kinematics problems, the computations are much simpler and the results are sufficiently accurate and precise for most purposes. 

Another commonly used idealisation is the assumption that the universe is homogeneous or isotropic at large scales. In other words, as we peer out into the farthest depths of space, we assume that matter and energy are evenly distributed. As I will show in the forthcoming essay, this assumption seems to be both theoretically and empirically false. And it seems that so-called "dark energy" is merely an artefact of this simplifying assumption. 

Many theories have fallen because of employing a simplifying assumption that distorts answers to make them unsatisfying. 

A "spherical cow in a vacuum" sounds funny, but a good approximation can simplify a problem just enough to make it tractable and still provide sufficient accuracy and precision for our purposes. It's not that we should never idealise a scenario or make simplifying assumptions. The fact is that we always do this. All physical theories involve starting assumptions. Rather, the argument is pragmatic. The extent to which we idealise problems is determined by the ability of the model to explain phenomena to the level of accuracy and precision that our questions require. 

For example, if our question is, "How do we get a satellite into orbit around the moon?" we have a classic "three-body" problem (with four bodies: Earth, moon, sun, and satellite). Such problems are mathematically very difficult to solve. So we have to idealise and simplify the problem. For example, we can decide to ignore the gravitational attraction caused by the satellite, which is real but tiny. We can assume that space is relatively flat throughout. We can note that relativistic effects are also real but tiny. We don't have to slavishly use the most complex explanation for everything. Given our starting assumptions, we can just use Newton's law of gravitation to calculate orbits. 

We got to relativity precisely because someone asked a question that Newtonian approaches could not explain, i.e. why does the orbit of Mercury precess and at what rate? In the Newton approximation, the orbit doesn't precess. But in Einstein's reformulation of gravity as the geometry of spacetime, a precession is expected and can be calculated. 

Models

I was in a physical chemistry class in 1986 when I realised that what I had been learning through school and university was a series of increasingly sophisticated models, and the latest model (quantum physics) was still a model. At no point did we get to reality. There did seem to me to be a reality beyond the models, but it seemed to be forever out of reach. I had next to no knowledge of philosophy at that point, so I struggled to articulate this thought, and I found it dispiriting. In writing this essay, I am completing a circle that I began as a naive 20-year-old student.

This intuition about science crystallised into the idea that no one has epistemic privilege. By this I mean that no one—gurus and scientists included—has privileged access to reality. Reality is inaccessible to everyone. No one knows the nature of reality or the extent of it. 

We all accumulate data via the same array of physical senses. That data feeds virtual models of world and self created by the brain. Those models both feed information to our first-person perspective, using the sensory apparatus of the brain to present images to our mind's eye. This means that what we "see" is at least two steps removed from reality. This limit applies to everyone, all the time.

However, when we compare notes on our experience, it's clear that some aspects of experience are independent of any individual observer (objective) and some of them are particular to individual observers (subjective). By focusing on and comparing notes about the objective aspects of experience, we can make reliable inferences about how the world works. This is what rescues metaphysics from positivism on one hand and superstition on the other. 

We can all make inferences from sense data. And we are able to make inferences that prove to be reliable guides to navigating the world and allow us to make satisfying causal explanations of phenomena. Science is an extension of this capacity, with added concern for accuracy, precision, and measurement error. 

Since reality is the same for everyone, valid models of reality should point in the same direction. Perhaps different approaches will highlight different aspects of reality, but we will be able to see how those aspects are related. This is generally the case for science. A theory about one aspect of reality has to be consistent, even compliant, with all the other aspects. Or if one theory is stubbornly out of sync, then that theory has to change, or all of science has to change. Famously, Einstein discovered several ways in which science had to change. For example, Einstein proved that time is particular rather than universal.  Every point in space has its own time. And this led to a general reconsideration of the role of time in our models and explanations. 

Sources of Error

A scientific measurement is always accompanied by an estimate of the error inherent in the measurement apparatus and procedure. Which gives us a nice heuristic: If a measurement you are looking at is not accompanied by an indication of the errors, then the measurement is either not scientific, or it has been decontextualised and, with the loss of this information, has been rendered effectively unscientific.

Part of every good scientific experiment is identifying sources of error and trying to eliminate or minimise them. For example, if I measure my height with three different rulers, will they all give the same answer? Perhaps I slumped a little on the second measurement? Perhaps the factory glitched, and one of the rulers is faulty? 

In practice, a measurement is accurate to some degree, precise to some degree, and contains inherent measurement error to some degree. And each degree should be specified to the extent that it is known.

Accuracy is itself a measurement, and as a quantity reflects how close to reality the measurement is. 

Precision represents how finely we are making distinctions in quantity.

Measurement error reflects uncertainty introduced into the measurement process by the apparatus and the procedure.

Now, precision is relatively easy to know and control. We often use the heuristic that a ruler is accurate to half the smallest measure. So a ruler marked with millimetres is considered precise to 0.5 mm. 

Let's I want to measure my tea cup. I have three different rulers. But I also note that the cup has rounded edges, so knowing where to measure from is a judgment call. I estimate that this will add a further 1 mm of error. Here are my results: 

• 83.5 ± 1.5 mm.
• 86.0 ± 1.5 mm.
• 84.5 ± 1.5 mm

The average is 84.6 ± 1.5 mm. So we would say that we think the true answer lies between 86.1 and 83.1 mm. And note that even though I have an outlier (86.0 mm), this is in fact within the margin of error. 

As I was measuring, I noted another potential source of error. I was guesstimating where the widest point was. And I think this probably adds another 1-2 mm of measurement error. When considering sources of error in a measurement, one's measurement procedure is often a source. In science, clearly stating one's procedure allows others to notice problems the scientists might have overlooked. Here, I might have decided to mark the cup so that I measured at the same point each time. 

Now the trick is that there is no way to get behind the measurement and check with reality. So, accuracy has to be defined pragmatically as well. One way is to rely on statistics. For example, one makes many measurements and presents the mean value and the standard deviation (which requires more than three measurements). 

The point is that error is always possible. It always has to be accounted for, preferably in advance. We can take steps to eliminate error. An approximation always relies on starting assumptions, and these are also a source of error. Keep in mind that this critique comes from scientists themselves. They haven't been blindly ignoring error all these years. 

Mathematical Models

I'm not going to dwell on this too much. But in science, our explanations and models usually take the form of an abstract symbolic mathematical equation. A simple, one-dimensional wave equation takes the general form:

y = f(x,t)

That is to say that the displacement of the wave (y) is a function of position (x) and time (t). Which is to say that changes in the displacement are proportional to changes in position in space and time. This describes a wave that, over time, moves in the x direction (left-right) and displaces in the y direction (up-down). 

More specifically, we model simple harmonic oscillations using the sine function. In this case, we know that spatial changes are a function of position and temporal changes are a function of time. 

y(x) = sin(x)
y(t) = sin(t)

It turns out that the relationship between the two functions can be expressed as 

y(x,t) = sin(x ± t).

If the wave is moving right, we subtract time, and if the wave is moving to the left, we add it. 

The sine function smoothly changes between +1 and -1, but a real wave has an amplitude, and we can scale the function by multiplying it by the amplitude.

y(x,t) = A sin(x ± t).

And so on. We keep refining the model until we get to the general formula:

y(x,t) = A sin(kx ± ωt ± ϕ).

Where A is the maximum amplitude, k is the stiffness of the waving medium, ω is the angular velocity, and ϕ is the phase.

The displacement is periodic in both space and time. Since k = 2π/λ (where λ is the wavelength), the function returns to the same spatial configuration when x/n = λ (where n is a whole number). Similarly, since ω = 2π/T (where T is the period or wavetime), the function returns to the same temporal configuration when t/n = T.

What distinguishes physics from pure maths is that, in physics, each term in an equation has a physical significance or interpretation. The maths aims to represent changes in our system over time and space. 

Of course, this is idealised. It's one-dimensional. Each oscillation is identical to the last. The model has no friction. If I add a term for friction, it will only be an approximation of what friction does. But no matter how many terms I add, the model is still a model. It's still an idealisation of the problem. And the answers it gives are still approximations.

Conclusion

No one has epistemic privilege. This means that all metaphysical views are speculative. However, we need not capitulate to solipsism (we can only rely on our own judgements), relativism (all knowledge has equal value) or positivism (no metaphysics is possible). 

Because, in some cases, we are speculating based on comparing notes about empirical data. This allows us to pragmatically define metaphysical terms like reality, space, time, and causality in such a way that our explanations provide us with reliable knowledge. That is to say, knowledge we can apply and get expected results. Every day I wake up and the physical parameters of the universe are the same, even if everything I see is different. 

Reality is the world of observer-independent phenomena. No matter who is looking, when we compare notes, we broadly agree on what we saw. There is no reason to infer that reality is perfect, absolute, or magical. It's not the case that somewhere out in the unknown, all of our problems will be solved. As a historian of religion, I recognise the urge to utopian thinking and I reject it. 

Rather, reality is seen to be consistent across observations and over time. Note that I say "consistent", not "the same". Reality is clearly changing all the time. But the changes we perceive follow patterns. And the patterns are consistent enough to be comprehensible. 

The motions of stars and planets are comprehensible: we can form explanations for these that satisfactorily answer the questions people ask. The patterns of weather are comprehensible even when unpredictable. People, on the other hand, remain incomprehensible to me.

That said, all answers to scientific questions are approximations, based on idealisations and assumptions. Which is fine if we make clear how we have idealised a situation and what assumptions we have made. This allows other people to critique our ideas and practices. As Mercier and Sperber point out, it's only in critique that humans actually use reasoning (An Argumentative Theory of Reason,10 May 2013). 

We can approximate reality, but we should not attempt to appropriate it by insisting that our approximations are reality. Our theories and mathematics are always the map, never the territory. The phenomenon may be real, but the maths never is.  

This means that if our theory doesn't fit reality (or the data), we should not change reality (or the data); we should change the theory. No mathematical approximation is so good that it demands that we redefine reality. Hence, all of the quantum Ψ-ontologies are bogus. The quantum wavefunction is a highly abstract concept; it is not real. For a deeper dive into this topic, see Chang (1997), which requires a working knowledge of how the quantum formalism works, but makes some extremely cogent points about idealised measurements.

In agreeing that the scientific method and scientific explanations have limits, I do not mean to dismiss them. Science is by far the most successful knowledge seeking enterprise in history. Science provides satisfactory answers many questions. For better or worse, science has transformed our lives (and the lives of every living thing on the planet). 

No, we don't get the kinds of answers that religion has long promised humanity. There is no certainty, we will never know the nature of reality, we still die, and so on. But then religion never had any good answers to these questions either. 

~~Φ~~

Beller, Mara. (1996). "The Rhetoric of Antirealism and the Copenhagen Spirit". Philosophy of Science 63(2): 183-204.

Chang, Hasok. (1997). "On the Applicability of the Quantum Measurement Formalism." Erkenntnis 46(2): 143-163. https://www.jstor.org/stable/20012757

Faye, Jan.(2007). "The Pragmatic-Rhetorical Theory of Explanation." In Rethinking Explanation. Boston Studies in the Philosophy of Science, 43-68. Edited by J. Persson and P. Yikoski. Dordrecht: Springer.

Lawson, E. T. and McCauley, R. N. (1990). Rethinking Religion: Connecting Cognition and Culture. Cambridge: Cambridge University Press.

Note: 14/6/25. The maths is deterministic, but does this mean that reality is deterministic? 

The Curious Case of Phlogiston

I'm fascinated by revisionist histories. I grew up in a British colony where we were systematically lied to about our own history. Events in the 1970s and 1980s forced us to begin to confront what really happened when we colonised New Zealand. At around the same time, modern histories began to appear to give us a more accurate account. James Belich's Making Peoples had a major impact on me. Michael King's Being Pakeha also struck a chord, as did Maurice Shadbolt's historical novel Monday's Warriors.

Most people who know a little bit about the history of science will have heard of phlogiston. The phlogiston theory is usually portrayed as exactly the kind of speculative metaphysics that was laid to rest by artful empiricists. Phlogiston became a symbol of the triumph of empiricism over superstition. As a student of chemistry, I imbibed this history and internalised it. 

The popular history (aka science folklore) has a Whiggish feel in the sense that Lavoisier is represented as making a rational leap towards the telos of the modern view. Such, we are led to believe, is the nature of scientific progress. My favourite encyclopedia repeats the standard folklore:

The phlogiston theory was discredited by Antoine Lavoisier between 1770 and 1790. He studied the gain or loss of weight when tin, lead, phosphorus, and sulfur underwent reactions of oxidation or reduction (deoxidation); and he showed that the newly discovered element oxygen was always involved. Although a number of chemists—notably Joseph Priestley, one of the discoverers of oxygen—tried to retain some form of the phlogiston theory, by 1800 practically every chemist recognized the correctness of Lavoisier’s oxygen theory.—Encyclopedia Britannica.

Compare this remark by Hasok Chang (2012b: time 19:00) in his inaugural lecture as Hans Rausing Professor of History and Philosophy of Science, at Cambridge University:

I became a pluralist about science because I could not honestly convince myself that the phlogiston theory was simply wrong or even genuinely inferior to Lavoisier's oxygen-based chemical theory.

When I was reading about the systematic misrepresentation of the work of J. J. Thomson and Ernest Rutherford in physics folklore, Chang's lecture came to mind. I discovered Chang 4 or 5 years ago and have long wanted to review his account of phlogiston, but was caught up in other projects. In this essay, I will finally explore the basis for Chang's scepticism about the accepted history of phlogiston. While I largely rely on his book, Chang pursued this theme in two earlier articles (2009, 2010).

Setting the Scene

The story largely takes place in the mid-late eighteenth century. The two principal figures are Joseph Priestley (1733 – 1804) and Antoine-Laurent de Lavoisier (1743 – 1794). 

A caveat is that while I focus on these two figures, the historical events involved dozens, if not hundreds, of scientists. Even in the 1700s, science was a communal and cooperative affair; a slow conversation amongst experts. My theme here is not "great men of history". My aim is to explore the historiography of science and reset my own beliefs. Chang's revisionist history of phlogiston is fascinating by itself, but I am intrigued by how Chang uses it as leverage in his promotion of pluralism in science. Priestley and Lavoisier are just two pegs to hang a story on. And both were, ultimately, wrong about chemistry. 

Chang (2012: 2-5) introduces Priestley at some length. He refers to him as "a paragon of eighteenth-century amateur science" who "never went near a university", while noting that he was also a preacher and a "political consultant" (from what I read, Priestley was really more of a commentator and pamphleteer). As a member of a Protestant dissenting church, Priestley was barred from holding any public office or working in fields such as law or medicine. In the 1700s, British universities were still primarily concerned with training priests for the Church of England. That said, Priestley was elected a fellow of the Royal Society in 1766, which at least gained him the ears of fellow scientists. Priestley is known for his work identifying different gases in atmospheric air. He first discovered "fixed air" (i.e. carbon-dioxide) and became a minor celebrity with his invention of carbonated water. He also discovered oxygen, more on this below. 

However, Chang provides no similar introduction to Lavoisier. Rather, Lavoisier appears in a piecemeal way as a foil to his main character, Priestley. The disparity seems to be rhetorical. Part of Chang's argument for plurality in science is that Priestley was on the right track and has been treated poorly by historians of science. By focusing primarily on Priestley and treating Lavoisier as secondary, Chang might be seen as rebalancing a biased story.

I'm not sure that this succeeds, because as a reviewer, I now want to introduce Lavoisier to my readers, and I have to rely on third-party sources to do that. Chang doesn't just leave the reader hanging; he misses an opportunity to put Lavoisier in context and to draw some obvious comparisons. That Priestley and Lavoisier inhabited very different worlds is apposite to any history of phlogiston.

Lavoisier was an aristoi who inherited a large fortune at the age of five (when his mother died). He attended the finest schools where he became fascinated by the sciences (such as they were at the time). This was followed by university studies, where Lavoisier qualified as a lawyer, though he never practised law (he did not need to). As an aristo, Lavoisier had access to the ruling elite, which gave him leverage in his dispute with Priestley. He was also something of a humanitarian and philanthropist, spending some of his fortune on such projects as clean drinking water, prison reform, and public education. Despite this, he was guillotined during the French Revolution after being accused of corruption in his role as a tax collector. He was later exonerated of corruption.

The contrasting social circumstances help to explain why Lavoisier was able to persuade scientists to abandon phlogiston for his oxygen theory. Lavoisier had money and class on his side in a world almost completely dominated by money and class. 

Having introduced the main players, we now need to backtrack a little to put their work in its historical context. In the 1700s, the Aristotelian idea that the world is made of earth, water, fire, and air was still widely believed. To be clear, both water and air were considered to be elemental substances. 18th-century medicine was still entirely rooted in this worldview.

Alchemy still fascinated the intelligentsia of the day. On one level, alchemists pursued mundane goals, such as turning lead into gold, and on another, they sought physical immortality (i.e. immortality in this life rather than in the afterlife).

The telescope and microscope were invented in the early 1600s. With the former, Galileo observed the Moon and Jupiter's satellites, becoming the first empirical scientist to upset the existing worldview by discovering new facts about the world. 

That worldview was still largely the synthesis of Christian doctrine with Aristotelian philosophy created by Thomas Aquinas (1225–1274). The microscope had also begun to reveal a level of structure to the world, and to life, that no one had previously suspected existed. The practice of alchemy began to give way to natural philosophy, i.e. the systematic investigation of properties of matter. Priestley and Lavoisier were not the only people doing this, by any means, but they were amongst the leading exponents of natural philosophy. 

One of the key phenomena that captured the attention of natural philosophers, for obvious reasons, was combustion. The ancient Greeks believed that fire was elemental and that combustion released the fire element latent in the fuel. This is the precursor to the idea of phlogiston as a substance.

Phlogiston Theory

The first attempt at a systematic account of phlogiston is generally credited to Georg Ernst Stahl (1659 – 1734) in Zymotechnia fundamentalis "Fundamentals of the Art of Fermentation" (1697). The term phlogiston derives from the Greek φλόξ phlóx "flame", and was already in use when it was applied to chemistry.

The basic idea was that anything which burns contains a mixture of ash and phlogiston. Combustion is the process by which phlogiston is expelled from matter, leaving behind ash. And we see this process happening in the form of flames. And thus, a combustible substance was one that contained phlogiston. Phlogiston was "the principle of inflammability". 

However, experimentation had begun to show interesting relationships between metals and metal-oxides (known at the time as calx). One could be turned into the other, and back again. For example, metallic iron gradually transforms into a reddish calx, which is a mixture of a couple of different oxides of iron. To turn iron-oxide back into iron, we mix it with charcoal or coke and heat it strongly. And this reversible reaction is common to all metals. 

Chemists used phlogiston to explain this phenomenon. Metals, they conjectured, were rich in phlogiston. This is why metals have such qualities as lustre, malleability, ductility, and electrical conductivity. In becoming a calx, the metal must be losing phlogiston, and by analogy, a calx is a kind of ash. On the other hand, charcoal and coke burn readily, so they must also be rich in phlogiston. When heated together, the phlogiston must move from the charcoal back into the calx, reconstituting the metal. 

This reversible reaction was striking enough for Immanuel Kant to use it, in The Critique of Pure Reason (1781), as an example of how science "began to grapple with nature in a principled way" (Chang 2012: 4).

Priestley is famous for having discovered oxygen, but as Chang emphasises, it was Lavoisier who called it that. Priestley called it dephlogisticated air, i.e. air from which phlogiston has been removed. "Air" in this context is the same as "gas" in modern parlance.

Priestley produced dephlogisticated air by heating mercury-oxide, releasing oxygen and leaving the pure metal. According to the phlogiston theory, such dephlogisticated air should readily support combustion because, being dephlogisticated, it would readily accept phlogiston from combustion. And so it proved. Combustion in dephlogisticated air was much more vigorous. Breathing the new air also made one feel invigorated. Priestley was the first human to breathe pure oxygen, though he tested it on a mouse first.

Formerly considered elemental, atmospherical air could now be divided into "fixed air" and "dephlogisticated air". A missing piece was inflammable air (hydrogen), which was discovered by Henry Cavendish in 1766, when he was observing the effects of acids on metals. Cavendish had also combusted dephlogisticated air and inflammable air to make water. And Priestley had replicated this in his own lab.

Priestley and Cavendish initially suspected that inflammable air was in fact phlogiston itself, driven from the metal by the action of acids. A calx in acid produced no inflammable air, because it was already dephlogisticated. However, the fact that dephlogisticated air and phlogiston combined to make water was suggestive and led to an important refinement of the phlogiston theory.

They settled on the idea that inflammable air (hydrogen) was phlogisticated water, and that dephlogisticated air (oxygen) was actually dephlogisticated water. And thus, the two airs combined to form water. In this view, water is still elemental. 

It was Lavoisier who correctly interpreted this reaction to mean that water was not an element but a compound of hydrogen and oxygen (and it was Lavoisier who named inflammable air hydrogen, i.e. "water maker"). However, it is precisely here, Chang argues that phlogiston proves itself to be the superior theory.

Chang notes that, without the benefit of hindsight, it's difficult to say what is so wrong with the phlogiston theory. It gave us a working explanation of certain chemical phenomena, and it made testable predictions that were accurate enough to be taken seriously. For its time, phlogiston was a perfectly good scientific theory. So the question then becomes, "Why do we see it as a characteristic example of a bad scientific theory disproved by empiricism?" Was it really such a bad theory?

A Scientific Blunder?

On one hand, Chang argues that, given the times, phlogiston theory was a step in the right direction, away from alchemical views and towards seeing electricity as the flow of a fluid, which then leads towards the modern view of chemical reactions involving the flow or exchange of electrons. And on the other hand, Lavoisier's theory is far from being "correct".

If the argument is that phlogiston was an ad hoc concept that could not be observed, then why is the same criticism not levelled against Lavoisier for the role of elemental luminaire or caloric in his theory? Caloric is what we would now call "heat", and it is clearly not elemental.

The terms "oxidation" and "reduction" (and the portmanteau "redox") are generalisations from Lavoisier's explanation of metals and metal-oxides. A metal-oxide can be "reduced" to the pure metal, and a metal oxidised to form the oxide. And one can make them go back and forth by altering the conditions.

While oxidation and reduction apply to metals and their oxides, such reactions are not typical. Most redox reactions don't involve metals or oxygen. When fluorine reacts with hydrogen, for example, we say that hydrogen is "oxidised" (gives up an electron) and that fluorine is "reduced" (gains an electron). And this terminology doesn't make much sense. Even with a BSc in chemistry, I always have to stop and think carefully about which label applies because it's not intuitive.

A commonly cited reason for the collapse of the phlogiston theory is that a metal gains weight in becoming a calx. The implication is that phlogiston theory was at a loss to explain this. Superficially, the early versions of phlogiston theory argue that in becoming a calx, the metal loses phlogiston, so we would expect it to lose weight, rather than gain it. The idea that the metal combines with oxygen is correct in hindsight, and is how we see the formation of metal-oxides in the present.

However, Priestley and another phlogistonist, Richard Kirwan, did have an explanation for weight gain. I've already noted that Priestley's ideas matured and that, latterly, he had concluded that inflammable air (hydrogen) was phlogisticated water, and dephlogisticated air (oxygen) was dephlogisticated water. In Priestley's mature view, the metal formed a calx by combination with water and the loss of phlogiston. The added weight was due to the dephlogisticated water. When the calx was reduced, the metal absorbed phlogiston and gave up water. 

Like Chang, when I review this explanation, keeping in mind the state of knowledge at the time, I can't see how Lavoisier's explanation is any better. Seen in the context of the times (late 18th century), there was nothing illogical about the phlogiston theory. It explained observations and made testable predictions. As Chang (2010: 50) says:

We really need to lose the habit of treating ‘phlogiston theory got X wrong’ as the end of the story; we also need to ask whether Lavoisier’s theory got X right, and whether it did not get Y and Z wrong.

Chang cites several historians of science commenting on this. For example, John McEvoy (1997) notes that...

by the end of the eighteenth century, almost every major theoretical claim that Lavoisier made about the nature and function of oxygen had been found wanting.

And Robert Siegfried (1988):

The central assumptions that had guided [Lavoisier's] work so fruitfully were proved empirically false by about 1815.

These comments are in striking contrast to the claim made by Britannica: "by 1800, practically every chemist recognized the correctness of Lavoisier’s oxygen theory". The story in Britannica is the widely accepted version of history. At the same time, Chang makes clear, the story in Britannica is simply false.

Lavoisier's theory of acids, his theory of combustion, and his theory of caloric were all clearly wrong from the viewpoint of modern chemistry. For example, Lavoisier claimed that all acids contain oxygen (the name oxygen means "acid maker"). However, hydrochloric acid (which we have in our stomachs) does not contain oxygen. Indeed, the action of acids is now thought to be because of their ability to produce hydrogen ions (aka naked protons, aka phlogisticated water), which are extremely reactive.

Moreover, as Chang (2012: 9) shows, the problems with Lavoisier's theory were well known to his contemporaries. Many scientists voiced their concerns at the time. The point is well taken. If we are judging by modern standards, then Lavoisier and Priestley were both wrong, Lavoisier no less than Priestley. Nonetheless, Lavoisier, with his fortune and his access to the French aristoi, had more leverage than dissenting Priestley.

That said, Lavoisier clearly won the argument. And the brief account of his triumph in Britannica is a classic example of the adage that the victors write history.

What We Lost

What Chang tries to do next is declared by the subtitle of section 2: "Why Phlogiston Should Have Lived" (2012: 14). The first section of the book is deliberately written relatively informally with the idea that a general reader could appreciate the argument. In this second section, however, he develops a much more philosophically rigorous approach and introduces a great deal more jargon, some of which is specific to his project.

My aim in this essay is to continue the discussion at the same level. This inevitably means losing exactly the nuances that Chang introduces and probably diverging from his intentions to some extent. I do recommend reading the rest of his argument. What follows is my, all too brief, interpretation of Chang's argument. 

While his history is revisionist, Chang's point is not to promote a speculative counterfactual history (which is to say, a fictitious alternative history). Rather, he seeks to make an argument for pluralism. Where pluralism means the coexistence of different explanations for any given phenomenon, until such time as the best explanation emerges. 

Chang argues that Lavoisier's view that oxygen was being exchanged in chemical reactions was clearly inferior and only applicable to metal/calx reactions. By the time this became clear, phlogiston was discredited and could not be revived. And Lavoisier's counterintuitive oxidation-reduction model became the norm in chemistry, and still is, despite its obvious disadvantages. 

The idea that phlogiston was being exchanged in chemical reactions was not a bad theory (for the time). Moreover, phlogiston was already conceptually linked to electricity. Getting from redox to the exchange of electrons took another century. Chang argues that the conceptual leap from phlogiston to the exchange of electrons could have been considerably easier than it was, starting from Lavoisier's theory.

Chang's argument for pluralism is not simply based on the two theories being equally false. Indeed, he goes to some pains to explain what they both got right. The point is that the phlogiston theory had untapped potential. In prematurely killing off phlogiston and adopting Lavoisier's oxygen theory (which as we have seen was disproved a few decades later), we actually retarded the progress of science. And when Lavoisier was proven wrong, we had no alternative theory and simply retained his awkward and misleading terminology. 

Had we allowed the two theories to co-exist a little longer, so that Lavoisier's explanation could be thoroughly tested and proven false before it was adopted, there is a possibility that we might have lighted on the electron exchange theory of chemical reactions a century earlier than we did. Indeed, as hinted above, phlogiston was already linked to electricity. Seen with hindsight, the rush to judgment about chemical reactions meant that scientists of the late 17th and early 18th centuries missed a huge opportunity. 

Chang is a pragmatist. He knows we cannot go back. His argument is that we should be alert to this situation in the present and the future and be less eager to settle on a theory where ambiguity remains. Arguably, the temporary triumph of the various Copenhagen interpretations of Schrödinger's equation was a similar example. We settled too early, for reasons unconnected to science, only to have the chosen theory be disproved some decades later. 

I don't read Chang as saying that we should hold on to pluralism no matter what. Only that, where there is room for doubt, we should allow multiple explanations to coexist, because we don't know in advance what the best answer will be. This only emerges over time. And a scientific theory can only benefit from responding to the challenges that other explanations pose.

Conclusions

Hasok Chang aims to demonstrate the value of pluralism through critiquing the history of the so-called "chemical revolution" identified with Lavoisier. And the case of phlogiston is both fascinating in its own right and a compelling study of how the lack of pluralism retarded the progress of science. 

While sources like Britannica follow science folklore in insisting on the "correctness" of the oxygen theory, historians of science tell us a different story. It may be true that Lavoisier's theory was widely adopted by 1800, but historians have shown that it was also largely falsified by 1815. By this time, the phlogiston theory had been "killed", as Chang puts it.

Chang attempts to show that phlogiston was not such a bad theory and that the oxygen theory was not such a good theory. Contrary to the usual Whiggish accounts, the triumph of Lavoisier's oxygen theory was not really an example of "scientific progress". Indeed, Chang supposes that adopting the oxygen theory actually retarded the progress of science, since it pointed away from the role of electricity in chemistry. This important insight took another century to emerge.

The phlogiston theory is arguably the better of the two theories that existed in the late 1700s. Chang argues that had phlogiston persisted just a little longer, at least until Lavoisier was disproved, we might have made the leap to seeing chemical reactions in terms of the flow of electricity between elements much earlier than we eventually did. And who knows what else this might have changed?

The point is not to inaugurate some kind of neo-phlogistonist movement or to speculate about counterfactual (alternative) histories. The point is that when we have competing theories, in the present, we should allow them to coexist rather than rushing to settle on one of them. 

Pluralism is a pragmatic approach to uncertainty. When different explanations are possible, we can compare and contrast the differences. Allowing such challenges is more likely to result in scientific progress than the rush to judgment or the overwhelming desire to have one right answer.

As noted at the outset, in this essay, I have largely overlooked the contributions of Priestley's and Lavoisier's contemporaries. I have emphasised the two main players, even more than Chang does, purely for narrative simplicity (and keeping this essay to a reasonable length). This might make it seem that it was something like a personal competition, when that doesn't seem to be the case. Think of this essay as a taster. My aim is to whet your appetite to go and discover Chang for yourself, or better, to go and read the original papers being published at the time. See for yourself.  

Coda

The pluralism that Chang praises in the case of chemistry is not the same kind of pluralism that exists in so-called "interpretations of quantum mechanics". Chang is in favour of having multiple explanations of a phenomenon until such time as the best explanation unequivocally emerges. But he also considers that the best explanations change over time as new data comes in. Chang is a pragmatist, and this seems to be the only viable approach to science. We do not and cannot acquire metaphysical certainty because there is no epistemic privilege with respect to reality. We are all inferring facts about reality based on experience, a procedure known to be fraught with difficulties that often go unnoticed.

Generally, in science, we see competing explanations that attempt to fit a new phenomenon into our pre-existing metaphysics. In crude terms, scientific theories are made to fit into existing views about reality, and new data changes our view of reality only rarely and often incrementally. Paradigms do change, but only with great reluctance. This conservatism is generally a good thing as long as it doesn't become dogmatic.

In stark contrast to the rest of science, in quantum physics, the mathematical approximations are considered infallible and inviolable, and scientists propose different realities in which the mathematics makes sense. They have become dogmatic about their theory and refuse to consider other models. It has not gone well.

As Sabine Hossenfelder said, "Theoretical physicists used to explain what was observed. Now they try to explain why they can’t explain what was not observed."

~~Φ~~

Bibliography

Chang, Hasok. (2009) "We Have Never Been Whiggish (About Phlogiston)". Centaurus 51(4): 239-264. https://doi.org/10.1111/j.1600-0498.2009.00150.x

Chang, Hasok. (2010). "The Hidden History of Phlogiston: How Philosophical Failure Can Generate Historiographical Refinement." HYLE – International Journal for Philosophy of Chemistry 16 (2): 47-79. Online.

Chang, Hasok. (2012a). Is Water H20? Evidence, Realism and Pluralism. Springer.

Chang, Hasok. (2012b). "Scientific Pluralism and the Mission of History and Philosophy of Science." Inaugural Lecture by Professor Hasok Chang, Hans Rausing Professor of History and Philosophy of Science, 11 October 2012. https://www.youtube.com/watch?v=zGUsIf9qYw8

Stahl, Georg Ernst. (1697). Zymotechnia fundamentalis.

Why Quantum Mechanics is Currently Wrong and How to Fix It.

It is now almost a century since "quantum mechanics" became established as the dominant paradigm for thinking about the structure and motion of matter on the nanoscale. And yet the one thing quantum mechanics cannot do is explain what it purports to describe. Sure, quantum mechanics can predict the probability of measurements. However, no one knows how it does this. 

Presently, no one understands the foundations of quantum mechanics. 

Feynman's quote to this effect is still accurate. It has recently been restated by David Deutsch, for example:

"So, I think that quantum theory is definitely false. I think that general relativity is definitely false." (t = 1:16:13)

"Certainly, both relativity and quantum theory are extremely good approximations in the situations where we want to apply them... So, yes, certainly, good approximations for practical purposes, but so is Newton's theory. That's also false." (t = 1:28:35)

—David Deutsch on Sean Carroll's podcast.

I listened to these striking comments again recently. This time around, I realised that my conception of quantum field theory (QFT) was entirely wrong. I have a realistic picture in my head, i.e. when I talk about "waves", something is waving. This is not what GFT says at all. The "fields" in question are entirely abstract. What is waving in quantum mechanics is the notion of the probability of a particle appearing at a certain location within the atom. Below I will show that this thinking is incoherent. 

There have been numerous attempts to reify the quantum wavefunction. And they all lead to ridiculous metaphysics. Some of the most hilarious metaphysics that quantum mechanics has produced are:

1. The universe behaves one way when we look at it, and a completely different way when we don't.
   
2. The entire universe is constantly, and instantaneously, splitting into multiple copies of itself, each located in exactly the same physical space, but with no connections between the copies.
   
3. Electrons are made of waves of probability that randomly collapse to make electrons into real particles for a moment.

None of these ideas is remotely compatible with any of the others. And far from there being a consensus, the gaps between "interpretations" are still widening. Anyone familiar with my work on the Heart Sutra will recognise this statement. It's exactly what I said about interpretations of the Heart Sutra.

Physics has lost its grip on reality. It has a schizoid ("splitting") disorder. I believe I know why.

What Went Wrong?

The standard quantum model embraces wave-particle duality as a fundamental postulate. In the 1920s, experiments seemed to confirm this. This is where the problems start.

Schiff's (1968) graduate-level textbook, Quantum Mechanics, discusses the idea that particles might be considered "wave packets":

The relation (1.2) between momentum and wavelength, which is known experimentally to be valid for both photons and particles, suggests that it might be possible to use concentrated bunches of waves to describe localized particles of matter and quanta of radiation. To fix our ideas, we shall consider a wave amplitude or wave function that depends on the space coordinates x, y, z and the time t. This quantity is assumed to have three basic properties. First, it can interfere with itself, so that it can account for the results of diffraction experiments. Second, it is large in magnitude where the particle or photon is likely to be and small elsewhere. And third, will be regarded as describing the behavior of a single particle or photon, not the statistical distribution of a number of such quanta. (Schiff 1968: 14-15. Emphasis added)

I think this statement exemplifies the schizoid nature of quantum mechanics. The Schrödinger model begins with a particle, described as a "wave packet", using the mathematics of waves. The problem is that physicists still want to use the wave equation to recover the "position" or "momentum" of the electron in the atom, as though it is a particle. I have seen people dispute that this was Schrödinger's intention, but it's certainly how Schiff saw it, and his text was widely respected in its day.

The obvious problem is that, having modelled the electron as a wave, how do we then extract from it information about particles, such as position and momentum? Mathematically, the two ideas are not compatible. Wave-talk and particle-talk cannot really co-exist. 

In fact, Schrödinger was at a loss to explain this. It was Max Born who pointed out that if you take the modulus squared value of the wave function (which outputs complex-numbered vectors), you get a probability distribution that allows you to predict measurements. As I understand it, Schrödinger did not like this at all. In an attempt to discredit this approach, he formulated his classic thought experiment of the cat in the box. A polemic that failed so badly, that the Copenhagen crowd adopted Schrödinger's cat as their mascot. I'll come back to this.

However, there is a caveat here. No one has ever measured the position of an electron in an atom, and no one ever will. It's not possible. We have probes that can map out forces around atoms, but we don't have a probe that we, say, can stick into an atom and wait for the electron to run into it. This is not how things work on this scale.

Can We Do Better? (Yes We Can!)

Electric charge is thought to be a fundamental property of matter. We visualise the electric charge of a proton as a field of electric potentials with a value at every point in space, whose amplitude drops off as the square of the distance. The electric field around a proton is observed to be symmetrical in three dimensions. In two dimensions, a proton looks something like this with radiating, evenly spaced field lines:

An electron looks the same, but the arrows point inwards (the directionality of charge is purely conventional). So if the electron were a point charge, an atom would be an electric dipole, like this:

This diagram shows that if the electron were a point mass/charge, the hydrogen atom would be subject to unbalanced forces. Such an atom would be unstable. Moreover, a moving electric dipole causes fluctuations in the magnetic field that would rapidly bleed energy away from the atom, so if it didn't collapse instantaneously, it would collapse rapidly. 

Observation shows atoms to be quite stable. So, at least in an atom, an electron cannot be a point mass/charge. And therefore, in an atom, an electron is not a point mass/charge.

Observation also shows that hydrogen atoms are electrically neutral. Given that the electric field of the proton is symmetrical in three dimensions, there is only one shape the electron could be and balance the electric charge. A sphere with the charge distributed evenly over it.

The average radius of the sphere would be the estimated value of the atomic radius. Around 53 picometers (0.053 nanometers) for hydrogen. The radius of a proton is estimated to be on the order of 1 femtometer.

Niels Bohr had a similar idea. He proposed that the electron formed a "cloud" around the nucleus. And this cloud was later identified as "a cloud of probability". Which is completely meaningless. The emperor is not wearing any clothes. As David Albert says on Sean Carroll's podcast:

“… there was just this long string of brilliant people who would spend an hour with Bohr, their entire lives would be changed. And one of the ways in which their lives were changed is that they were spouting gibberish that was completely beneath them about the foundations of quantum mechanics for the rest of their lives…” (emphasis added)

We can do better, with some simple logic. We begin by postulating, along with GFT, that the electron is some kind of wave. 

If the electron is a wave, AND the electron is a sphere, AND the atom is stable, AND the atom is electrically neutral, then the electron can only be a spherical standing wave.

Now, some people may say, "But this is exactly what Schrödinger said". Almost. There is a crucial difference. In this model, the spherical standing wave is the electron. Or, looked at from the other direction, an electron (in a hydrogen atom) is a physical sphere with an average radius of ~53 pm. There is no particle, we've logically ruled out particles.

What does observation tell us about the shape of atoms? We have some quite recent data on this. For example, as reported by Lisa Grossman (2013) for New Scientist, here are some pictures of a hydrogen atom recently created by experimenters.

The original paper was in Physical Review.

Sadly, the commentary provided by Grossman is the usual nonsense. But just look at these pictures. The atom is clearly a sphere in reality, just as I predicted using simple logic. Many crafty experiments, have reported the same result. It's not just that the probability function is spherical. Atoms are spheres. Not solid spheres, by any means, but spheres nonetheless.

We begin to part ways with the old boys. And we are instantly in almost virgin territory. To the best of my knowledge, no one has ever considered this scenario before (I've been searching the literature).

The standard line is that the last input classical physics had was Rutherford's planetary model proposed in 1911, after he successfully identified that atoms have a nucleus, which contains most of the mass of the atom. This model was debunked by Bohr in 1913. And classical physics has nothing more to say. As far as any seems to know, "classical physics says the electron is a point mass". No one has ever modelled the electron in an atom as a real wave. At least no one I can find.

This means that there are no existing mathematical models I can adapt to my purpose. I have to start with the general wave equation and customise it to fit. Here is the generalised wave equation of a spherical standing wave:

Where r is the radius of the sphere, θ and φ are angles, and t = time. Notice that it is a second-order partial differential equation, and that the rates of change in each quantity are interdependent. It can be solved, but it is not easy.

The fact is that, while this approach is not identical to existing quantum formalism, it is isomorphic (i.e. has the same form). Once we clarify the concept and what we are trying to do with it, the existing formalism ought to be able to be adapted. So we don't have to abandon quantum mechanics, we just have to alter our starting assumptions and allow that to work through what we have to date. 

An important question arises: What about the whole idea of wave-particle duality?

In my view, any particle-like behaviour is a consequence of experimental design. Sticking with electrons, we may say that every electron detector relies on atoms in the detector absorbing electrons. And there are no fractional electrons. Each electron is absorbed by one and only one atom. It is this phenomenon that causes the appearance of discrete "particle-like" behaviour. At the nano-scale, any scientific apparatus is inevitably an active part of the system.

The electron is a wave. It is not a particle. 

Given the wild success of quantum mechanics (electronics, lasers, and so on), why would anyone want to debunk it? For me, it is because it doesn't explain anything. I didn't get into science so I could predict measurements, by solving abstract maths problems. I got into it so I could understand the world. Inj physics maths is supposed to represent the world and to have a physical interpretation. I'm not ready to give up on that.

The Advantages of Modelling the Electron as a (Real) Wave.

While they are sometimes reported as special features of quantum systems, the fact is that all standing waves have some characteristic features.

In all standing waves, energy is quantised. This is because a standing wave only allows whole numbers of wavelengths. We may use the example of a guitar string that vibrates in one dimension*.

*Note that if you look at a real guitar string, you will see that it vibrates in two dimensions: perpendicular to the face of the guitar and parallel to it.

The ends of the string are anchored. So the amplitude of any wave is always zero at the ends; they cannot move at all. The lowest possible frequency is when the wavelength equals the string length.

The next lowest possible frequency is when the wavelength equals half the string length. And so on.

This generalises. All standing waves are quantised in this way. This is "the music of the spheres". 

Now, spherical standing waves, with a central attractive force exist and were described ca 1782 by Pierre-Simon Laplace. These entities are mathematically very much more complicated than a string vibrating in one dimension. Modelling this is a huge challenge. 

For the purposes of this essay, we can skip to the end and show you what the general case of harmonics of a spherical standing wave looks like when the equations are solved and plotted on a graph.

Anyone familiar with physical chemistry will find these generalised shapes familiar. These are the theoretical shapes of electron orbitals for hydrogen. And this is without any attempt to account for the particular situation of an electron in an atom (the coulomb potential, the electric field interfering with itself, etc).

So not only is the sphere representing the electron naturally quantised, but the harmonics give us electron "orbitals". And, if we drop the idea of the electron as a particle, this all comes from within a classical framework (though not Rutherford's classical framework). 

Why Does Attempting to Reify Probability Lead to Chaos?

As already noted, Schrödinger tried and failed to relate his equation back to reality. Max Born discovered that the modulus squared of the wavefunction vector at a given point could be interpreted as the probability of finding the "the electron" (qua particle) at that point. This accurately predicts the probable behaviour of an electron, though not its actual behaviour. But all this requires electrons to be both waves and point-mass particles. 

Since the real oscillations I'm describing are isomorphic with the notional oscillations predicted by Schrödinger, we can intuit that if we were to try to quantify the probability of the amplitude of the (real) spherical standing wave at a certain point around the sphere, then any probability distribution we created from this would also be isomorphic with application of the Born rule to Schrödinger's equation.

What I've just done, in case it wasn't obvious, is explain the fundamentals of quantum mechanics (in philosophical terms at least) in one sentence. The predicted probabilities take the form that they do because of a physical mechanism: a spherical standing wave. And I have not done any violence to the notion of "reality" in the process. To my knowledge, this has not been done before, although I'm certainly eager to learn if it has.

However, the isomorphism is only causal in one direction. You can never get from a probability distribution to a physical description. Let me explain why by using a simple analogy that can be generalised.

Let's take the very familiar and simple case of a system in which I toss a coin in the air and, when it lands, I note which face is up. The two possible outcomes are heads H and tails T. The probabilities are well-known:

P(H) = 0.5 and P(T) = 0.5.

And as always, the sum of the probabilities of all the outcomes is 1.0. So:

P(H) + P(T) = 1.0

No matter what values we assign to P(H) and P(T), they have to add up to 1.

In physical terms, this means that if we toss 100 coins, we expect to observe heads 50 times and tails 50 times. In practice, we will most likely not get exactly 50 of each because probabilities do not determine outcomes. Still, the more times we toss the coins, the closer our actual distribution will come to the expected value.

Now imagine that I have tossed a coin, it has landed, but I have not yet observed it (call this the one-dimensional Schrödinger's cat, if you like). The standard rhetoric is to say that the coin is in a superposition of two "states". One has to be very wary of the term "state" in this context. Quantum physicists do not use it in the normal way, and it can be very confusing. But I am going to use "state" in a completely naturalistic way. The "state" of the tossed coin refers to which face is up. And it has to be in one of two possible states: H or T.  

Now let's ask what I know and think about what I can know about the coin at this moment before I observe the state of the coin.

I know that the outcome must be H or T. And I know that the odds are 50:50 that it is either one. What else can I know? Nothing. Despite knowing to 100 decimal places what the probability is, I cannot use that information to know what state the coin is in before I observe it. If I start with probabilities, I can say nothing about the fact of the matter (using a phrase David Albert uses a lot). If I reify this concept, I might be tempted to say that there is no fact of the matter. 

Note also that it doesn't matter if P(H) and P(T) are changing. Let us say that the probabilities change over time and that the change can be precisely described by a function of the coin: Ψ(coin). Are we any better off? Clearly not.

This analogy generalises. No matter how complex my statistical model, no matter how accurately and precisely I know the probability distribution, I still cannot tell you which side up the coin is without looking. There is undoubtedly a physical fact of the matter, but as the old joke goes, you cannot get there from here.

There are an infinite number of reasons why a coin toss will have P(H) = P(T) = 0.5. We can speculate endlessly. This is why the "interpretations" of quantum mechanics are so wildly variable and the resulting metaphysics so counter-intuitive. Such speculations are not bound by the laws of nature. In fact, all such speculations propose radical new laws of nature, like splitting the entire universe in two every time a quantum event happens. 

So the whole project of trying to extract meaningful metaphysics from a probability distribution was wrong-headed from the start. It cannot work, and it does not work. A century of effort by very smart people has not produced any workable ideas. Or any consensus on how to find a workable idea. 

Superposition and the Measurement Problem

The infamous cat experiment, in all its varieties, involves a logical error. As much as Schrödinger resisted the idea, because of his assumption about wave-particle duality, his equation only tells us about the probabilities of states; it does not and cannot tell us which state happens to be the fact of the matter. The information we get from the current formalism is a probability distribution. So the superposition in question is only a superposition of probabilities; it's emphatically not a superposition of states (in my sense). A coin cannot ever be both H and T. That state is not a possible state. 

Is the superposition of probabilities in any way weird? Nope.

The fact that P(H) = 0.5 or P(H) = Ψ(coin) and that P(T) = 0.5 or P(T) = Ψ(coin) are not weird facts. Nor is the fact that P(H) + P(T) = 1. These are common or garden facts, with no mystical implications.

If we grant that the propositions P(H) = 0.5 and P(T) = 0.5 are logically true, then it must also be logically (and mathematically) true to say that P(H) + P(T) = 1. Prior to observations all probabilities coexist at the same time.

For all systems we might meet, all the probabilities for all the outcomes always coexist prior to observing the state of the system. And the probabilities for all but one outcome collapse to zero at the moment we observe the actual state. This is true for any system: coins, cats, electrons, and everything. 

Note also that this is not a collapse of anything physical. No attempt to reify this "collapse" should be made. Probability is an idea we can quantify, but it's not an entity. No existing thing collapses when we observe an event. 

Moreover, Buddhists and hippies take note, our observing an event cannot influence the outcome. Light from the event can only enter our eye after the event has occurred, i.e. only after the probabilities have collapsed. And it takes the brain an appreciable amount of time to register the incoming nerve signal, make sense of it, and present it to the first-person perspective. Observation is always retrospective. So no, observation cannot possibly play any role in determining outcomes. 

One has to remember that probability is abstract. It's an idea about how to quantify uncertainty. Probability is not inherent in nature; it comes from our side of the subject-object divide. Unlike, say, mass or charge, probability is not what a reductionist would call "fundamental". We discover probabilities through observation of long-term trends. At the risk of flogging a dead horse, you cannot start with an abstraction and extract from it a credible metaphysics. Not in the world that we live in. And after a century of trying, the best minds in physics have signally failed in this quixotic endeavour. There is not even a working theory of how to make metaphysics from probabilities. 

The superposition or collapse of probabilities is in no way weird. And this is the only superposition predicted by quantum mechanics. 

In my model, the electron is a wave, and the wave equation that describes it applies at all times. Before, during, and after observation. 

In my model, probabilities superpose when we don't know the facts of the matter, in a completely normal way. It's just that I admit the abstract nature of probability distributions. And I don't try to break reality so that I can reify an abstraction.

On the other hand, my approach is technically classical. A classical approach that ought to predict all the important observations of quantum mechanics, but which can also explain them in physical terms. As such, there is no separation between classical and quantum in my model. It's all classical. And I believe that the implications of this will turn out to be far-reaching and will allow many other inexplicable phenomena to be easily explained.

The so-called measurement problem can be seen as a product of misguided attempts to hypostatise and reify the quantum wavefunction, which only predicts probabilities. It was only ever a problem caused by a faulty conceptualisation of the problem in terms of wave-particle duality. If we drop this obviously false axiom, things will go a lot more smoothly (though the maths is still quite fiendish).

No one ever has or ever will observe a physical superposition. I'm saying that this is because no such thing exists or could exist. It's just nonsense, and we should be brave enough to stand up and say so.

There is no "measurement problem". There's measurement and there is ill-advised metaphysical speculation based on reified abstractions.

What about other quantum weirdness?

I want to keep this essay to a manageable length, so my answer to this question must wait. But I believe that Peter Jackson's (2013) free electron model as a vortex rotating on three axes is perfectly consistent with what I outlined here. And it explains spin very elegantly. If the electron is a sphere in an atom, why not allow it to always be a sphere?

Jackson also elegantly explains why the polarised filter set-up to test Bell's inequalities is not quantum weirdness, but a result of the photon interacting with, and thus being changed by, the filter. At the nano-scale and below, there are no neutral experimental apparatus.

What about interference and the double-slit experiment? Yep, I have some ideas on this as well.

Tunnelling? I confess that I have not tried to account for tunneling just yet. At face value, I think it is likely to turn out to be a case of absorption and re-emission (like Newton's cradle) rather than Star Trek-style teleporting. Again, there is no such thing as a neutral apparatus on the nano-scale or below. If your scientific apparatus is made of matter, it is an active participant in the experiment and at the nano-scale, it changes the outcomes. 

It's time to call bullshit on quantum mechanics and rescue physicists from themselves. After a century of bad metaphysics, let's put the phys back into physics!

~~Φ~~

P.S. My book on the Heart Sutra is coming along. I have a (non-committal) expression of interest from my publisher of choice. I hope to have news to share before the end of 2025.

PPS. I'd quite like to meet a maths genius with some time on their hands...

PPPS (16 Apr). I now have an answer to the question "What is waving?". An essay on this is in progress but may take a while. 

Bibliography

Grossman, Lisa. (2013). "Smile, hydrogen atom, you're on quantum camera." New Scientist. https://www.newscientist.com/article/mg21829194-900-smile-hydrogen-atom-youre-on-quantum-camera/

Jackson, Peter. (2009). "Ridiculous Simplicity". FQXi. What is Fundamental? https://forums.fqxi.org/d/495-perfect-symmetry-by-peter-a-jackson

Schiff, Leonard I. (1968). Quantum Mechanics. 3rd Ed. McGraw-Hill.

Classical is Cooler

Many extravagant claims are made for quantum physics, and in comparison classical physics often seems to be dismissed, almost as though it is of little consequence.

Amongst other things, it has long bugged me that Buddhists hijack quantum mechanics and combine it with the worst of Buddhist philosophy—i.e. Madhyamaka—to create a monstrous form of bullshit. I've previously written three essays that try to address the perennial quantum bullshit that thrives amongst Buddhists.

• Erwin Schrödinger Didn't Have a Cat (29 October 2010).
• Buddhism and the Observer Effect in Quantum Mechanics (18 July 2014)
• Quantum Bullshit (05 October 2018)

Although, I don't seem to have had any appreciable effect on the levels of bullshit.

In this essay, I'm going to make an argument that classical physics is, in fact, much cooler than quantum physics, especially the bullshit quantum physics that doesn't use any mathematics

Life, the Universe, and Everything.

One way of describing the observable universe is to state the scales of mass, length, and energy it covers.

• The total mass of the observable universe is thought to be in order or 10⁵³ kg. From the smallest objects (electrons) to the whole universe is about 84 orders of magnitude (powers of ten).
• The observable universe is about 4 x 10²⁶ metres in diameter; and from the smallest possible length (the Planck length) to the whole is about 61 orders of magnitude.
• E=mc² gives the total energy of the universe as about 10⁷⁰ joules, and covers about 61 orders of magnitude.

Human beings can perceive roughly 18 orders of magnitude of mass, 12 of length, and 11 of energy, roughly in the middle of each scale. Much of the universe is imperceptible to our naked senses. Human beings evolved and thrived for hundreds of thousands of years without knowing anything beyond what we could see, hear, smell, taste, or touch with our naked senses.

It was the invention of the ground glass lens that alerted us to the existence of both larger scales (telescope) and smaller scales (microscope). And for this reason I count the lens the most significant invention in the history of science. I know people count Copernicus as the first European scientist, but to my mind he was merely a precursor. Galileo was the first to make systematic observations and thereby discover new things about the universe, e.g. acceleration due to gravity is a constant, the moon's surface is not smooth but cratered, and that Jupiter has satellites. Note that Galileo did not have evidence or a good case for a "heliocentric universe" (and his ideas about this were wrong in several ways, but that's another story).

400 years later, we have a number of hugely successful theories of how the universe works. We've identified four fundamental forces and two kinds of particle: fermions and bosons. However, no single approach to physics can cover all the many orders of magnitude. All of our explanations are limited in their scope. Newtonian mechanics fails with large masses or high relative velocities. Relativity fails on the nanoscale and especially at the time of the big bang. Quantum physics fails on the macro-scale.

Physicists still hope to find a way of reconciling relativity and quantum physics, which they predict will produce a single mathematical formalism that can describe our universe at any scale. After more than a century of trying, we don't seem to be any closer to this. To be fair a lot of time, effort, and resources went into pursuing so-called "string theory" which has proven to be a dead end, at least as far as reconciling nano and macro physics. 

What I want to do in the rest of this essay is contrast classical physics and quantum physics.

Classical Physics

Classical physics is a primarily a description of the world that we perceive. As such, classical physics will always be salient and applicable to our lives. When we need a lever to move something, we use classical physics. When we want to describe the universe on the largest scale, we use classical physics. This means that classical physics is largely intuitive (even if the maths is not). 

Classical physics is testable and has been extensively tested. While it was never my favourite subject, I studied physics as a distinct subject for four years up to undergraduate level and in that time I did many experiments. I was able, for example, to observe the applicability of ideas like Newton's laws of motion. 

I have personally observed that m₁v₁ = m₂v₂ (i.e. momentum is conserved). And you can too, if you put your mind to it. Classical physics is highly democratic in the sense that anyone can test its predictions relatively easily.

Classical physics shows that the universe (on this scale) follows relatively simple patterns of evolution over time that can be written down as mathematical statements. In the 19th century, such expressions were called "laws". By the mid 20th century we called them "theories". Simple examples include:

• the relationship between pressure (P), volume (V), and temperature (T) of any gas is PV/T = constant.
• the relationship between voltage (V), current (I), and resistance (R) in a circuit is V=IR.
• the relationship between force and acceleration of an object with mass is F=ma.

The mathematics of relativity is considerably more complex than these examples, but one gains several degrees of accuracy (≈ numbers after the decimal point) as compensation.

An interesting feature of our experience of the world is that time goes in one direction. This is a consequence of entropy. We can always tell when a film is playing backwards, for example, because the causality is all wrong. Broken cups never spontaneously reform and leap up from the floor to appear unbroken in our hands. Whole cups common fall down to the floor and smash. Once again, classical physics is intuitive.

Classical physics has never been made into an analogy by New Age gurus. No one ever compared the Heart Sutra to classical physics. No one ever says classical physics is "weird" or counter-intuitive. The fixed speed of light is a little counter-intuitive but it doesn't lend itself to the kind of Romantic flights of fancy that make religion seem interesting. If anything, religieux are apt to dismiss the whole topic of classical physics as irrelevant to "spirituality". Classical physics seems to resist being co-opted by woo-mungers.

And then there is quantum physics...

Quantum

Mathematically, quantum physics is profoundly accurate and precise method of predicting probabilities. However, unlike classical physics no one knows why it works. Literally, no one knows how the mathematics relates to reality. There are lots of ideas, each more counter-intuitive than the next and each relies on a series of assumptions that are beyond the scope of the mathematical formalism. But each set of assumptions leads to radically different metaphysics! And there is no agreement on which assumptions are valid. And at present there is no way to test these theories. I've seen Sean Carroll argue that Many Worlds does make testable predictions, but as far as I know, they have not been tested.

Einstein was of the opinion that quantum physics was incomplete. Sadly his proposed solution to this seems to have been ruled out. But still, I think the only viable stance is to consider quantum theory as incomplete until such time as we know how it relates to reality.

Which brings us to the first false claim that is commonly asserted by scientists: "the universe is deterministic." This assumes that quantum theory explains how matter behaves. But it doesn't. We don't know how mathematics relates to reality. So we don't know if the universe is deterministic or not. The claim that the the universe is deterministic goes far beyond our present state of knowledge. Most interpretations of quantum physics treat it as probabilistic rather than deterministic. And this undermines all claims that the universe is deterministic.

Another common falsehood is "quantum mechanics is a description of reality". But it should already be apparent that this is simply not true. Physicists do not know how the mathematics of quantum physics relates to reality. All they know is that the mathematics accurately assesses the probabilities of the various states that the system can be in over time. It doesn't tell us what will happen, at best it tells us what can happen.

At the popular level, quantum physics is plagued by vagueness and misleading statements. Scientists talk about "the wavefunction" as an independent thing (hypostatisation) and even as a physical thing (reification), when is it in fact an abstract mathematical function. They talk about "wave-like" behaviour without ever distinguishing this from actual wave behaviour. "Observation", so crucial to some approaches, is vague and more or less impossible to define.

We see statements like "energy is quantised" as though all energy is quantised. But this is not true. If you measure radiation from the sun, for example, it smoothly spans the entire electromagnetic spectrum (the sun glows because its hot, and that glow is blackbody radiation which is smooth rather than discreet). Energy is only quantised in atoms. And the solar spectrum is itself proof of this because the atoms in the sun absorb energy at precise wavelengths, causing the spectrum of sunlight to have darker bands when viewed at a fine enough grain.

The quantisation in atoms is explained in terms of an electron in an atom being conceived of as a standing wave - which means it can only vibrate at frequencies that allow for a whole number of wavelengths. For example, the harmonic series on a guitar string is also "quantised": the diagram shows different modes of vibration. The top shows wavelength = string length. but the string can also vibrate at twice the fundamental frequency so that 2 wavelengths = string length, then 3, 4, 5, 6, and 7 wavelengths = string length (out to infinity).

The energy levels for electrons in atoms show a similar pattern. But remember that an electron is 3 dimensional. Spherical harmonics look more like this

Which is similar to how we think electron orbitals look in Hydrogen.

Some of these results are confirmed by the shapes of molecules, which can be determined independently, for example by X-ray crystallography.

People talk about "measuring where the electron is in the atom". But this is almost pure bullshit. No one has ever measured the position of an electron in an atom. It's not possible. Within an atom, an electron is distorted into a spherical standing wave. "Position" is meaningless in this context. As are most other particle-related ideas. And remember, we cannot solve the equations when there are two or more electrons, we can only estimate (though current estimates are still very accurate).

We also see statements like "a system can exist in multiple states simultaneously", usually referred to as superposition (the "position" part is entirely misleading). This phrase is often used in popular explanations of quantum mechanics, but it’s misleading. The wavefunction describes a superposition of probability amplitudes, it does not describe a coexistence of multiple physical states. In fact, the term "state"—as it is usually used—is not applicable here at all, precisely because in normal usage it implies existence. In this context "state" confusingly means every single possible state, each with its own probability.

For example, if an electron has the wavefunction is ψ = ψ₁ + ψ₂ it doesn’t mean the electron is "in both states ψ₁ and ψ₂ at once." This is because neither ψ₁ nor ψ₂ is a physical state. Each is a probability distribution. So what superposition means is that, at some time, the electron's state has a probability distribution that reflects the combined amplitudes of ψ₁ and ψ₂. There is and can be no superposition of physical states, nor is their any theoretical possibility of observing such a thing.

All of those "interpretations" that treat the wavefunction as real simply assert its existence as axiomatic and introduce further a priori assumptions into order to try to make sense of this mess. If we make no assumptions then there is nothing about the mathematical formalism of quantum mechanics that forces us to think of the wavefunction as a real thing rather than an abstraction. It's a probability distribution. Which is an abstraction.

Which means that the idea that the wave-function can "collapse" is nonsensical. All probability distributions without exception "collapse" at the point of measurement.

If I roll a die, I get one number facing up. It can be any one of the six numbers. And each number is equally likely to be face up after a roll. Before I roll the die, the "wavefunction" of the die describes 6 possible "states" each of which is equally likely. When I roll the die I get one answer. Has anything mysterious happened? I think not. Let's say I roll a 2. I don't have to explain what happened to 1,3,4,5 and 6. Nothing happened to them, because they are not things. They are just unrealised possibilities. I get one result because only one result is physically possible. But before I know which result I have, all the possibilities have a finite probability. There's nothing "weird" or "mysterious" about this unless one first reifies the wavefunction.

Indeed, the whole idea of the "measurement problem" appears to be based on a serious misconception (as far as I can see). The measurement problem is based on the idea that the Schrödinger equation describes a system as existing in multiple physical states. But it doesn't. It describes probability distribution of possible physical states. A potentiality is not an existing state.

The only time measuring becomes problematic is when we assume that the wavefunction is a thing (reification) or that it reflects existent states rather then potential states. And these moves are simply mistakes.

Ironically, the one thing that Schrödinger's equation is not, as Nick Lucid explains, is a wave equation. The generalised wave equation contains a second-order partial differential with respect to time (a distorting force is countered by a restoring force, causing acceleration). This is a fascinating observation. I gather that using the constant i (√-1) in the Schrödinger equation allows for some "wave-like" behaviour, but no one really talks about this in lectures on quantum physics. Nor do they distinguish "wave" from "wave-like". And we still have to insist that the "wave-like" behaviour in question is a wave of probability, not a physical wave.

But then Nick Lucid, who typically is quite lucid (despite his "crazy" schtick), also introduces his video by saying "Schrödinger's equation governs the behavior of tiny quantum particles by treating them as wave functions." No equation anywhere "governs" anything. The equation describes the probability of a range of possible states. It's a descriptive law, not a prescriptive law. And as Lucid goes on to say, the equation in question is not a wave equation, it's a heat equation. The one thing that Schrödinger's equation doesn't do is "govern the behavior of tiny quantum particles".

This generalises: physics is a description, not a prescription. Abstract mathematical expressions cannot "govern" concrete entities. And in the case of quantum physics, it doesn't seem to relate to the "behaviour" either, since it only predicts the probability of any given state following from the present state. So it's not even a description of actual behaviour, just a description of potential behaviour at any point in time. With the most precise prediction as to probability, we still don't know what's going to happen next, and the actual outcome could always be the least likely outcome. That's why quantum tunneling is a thing, for example.

Unlike classical physics, which every undergraduate students proves to their own satisfaction, nano-scale physics is impossible to observe directly. It takes massive, complicated, and expensive equipment to get information from that scale. Information goes through many stages of amplification and transformation (from one kind of energy to another) before anything perceptible emerges. And that has to be processed by powerful computers before it makes any sense. And then interpreted by human beings.

That blip on the graph at 125 GeV that the LHC produced as evidence of the Higgs Boson is abstracted to the n^th degree from the thing itself.

At no time was a Higgs Boson ever observed, and at no time in the future will one ever be observed. What was observed was a particular kind of decay product, which the logic of the standard model says can only be produced if a Higgs Boson decays in the way that Peter Higgs predicted. Assuming that the standard model is right. Keep in mind that the model didn't predict the energy of the Higgs particle exactly. There was actually a lot of uncertainty. And the two different detectors actually measured slightly different numbers. Moreover, do you see how wide that peak was? That width is experiment error. Maybe the energy of the Higgs is 125 GeV, or maybe its a little more or a little less?

We cannot ever see the nano-scale. And because of this, we simply cannot imagine the nano-scale.

A 1 gram diamond, for example contains in the order of 5 x 10²² atoms. How big would that diamond be if each atom of carbon was 1mm³ or roughly the size of a grain of salt? It would be 5 x 10¹³ cubic metres. This is roughly the volume of Mount Everest. So an atom is to a grain of salt, as a grain of salt is to Mt Everest.

Imagination simply fails.

Conclusion

In short, at least at the popular level, quantum physics is a constant source of vague or misleading information. It is plagued by careless use of language and outright false claims by scientists themselves. The philosophy of quantum physics is difficult, but on the whole it fails to adequately distinguish epistemology and metaphysics. This is made worse by kooks and charlatans leveraging the confusion to pull the wool over our eyes. Sometimes, the kooks and the scientists are in a superposition: notably Eugene Wigner's theory about "consciousness" (another abstraction) collapsing the wavefunction. Wigner won a Nobel, but he was also a serious kook. And he has been responsible for a mountain of bullshit as a result.

Most of what is said about quantum physics outside of university lecture halls is bullshit, and quite a bit that is said in them is also bullshit or at least partially digested hay. Everything that is said about Buddhism and quantum physics is mendacious bullshit.

There is no doubt that insights gained from quantum physics are important and valuable, but the whole thing is over-hyped and plagued by nonsense. The actual work is largely about approximating solutions to the insoluble mathematical equations, which at best give us probabilities. It works remarkably well, but no one knows why.

The idea that quantum physics is any kind of "description of reality" is pure bullshit. It's a probability distribution, for a reality that no understands any better now than when physics genius Richard Feynman said: "No one understands quantum mechanics".

Classical physics on the other hand is seldom vague or misleading. It resists being leveraged by kooks by being precisely and accurately defined. It can readily be tested by more or less anyone. Classical physics is much less prone to bullshit. No one ever bothers to compare Buddhism to classical physics. Which is a good sign.

Classical physics is not only cooler than quantum physics. It is way cooler. 

Coda

If anyone is still unconvinced that quantum theory has no conceivable relationship with Buddhism, then I invite you to watch this video introduction to quantum mechanics from an Oxford University undergraduate physics course. This is a no bullshit course. 

I defy anyone to connect anything said in this video to any aspect of Buddhist doctrine.