Roger Penrose on Kurt Gödel and Gödelian Truth

Physicist and mathematician Roger Penrose believes that the ascertaining of (what can be called) Gödelian truths is a non-algorithmic process. Indeed, he believes that “genuine intelligence” is non-algorithmic too. Kurt Gödel himself believed that intuition allows us to “grasp” various concepts. Like Descartes’ position on “clear and distinct ideas”, Gödel believed that if we grasp a concept (at least one of a limited, not always mathematical, type) in a clear way, then that will guarantee us knowledge of its truth.

Roger Penrose and Kurt Gödel.

Physicist and mathematician Roger Penrose believes in what can be called Gödelian truth. His belief can be placed in a larger context and it also has a history. That context and history is partly (or even largely) down to Penrose’s stance against mathematical formalism.

Here’s Penrose on such formalism:

“The point of view that one can dispense with the meanings of mathematical statements, regarding them as nothing but strings of symbols in some mathematical system, is the mathematical standpoint of formalism.”

Penrose has a serious problem with this “point of view”. He goes on to say that “[s]ome people like this idea, whereby mathematics becomes a kind of meaningless game”. Penrose concludes:

“It is not an idea that appeals to me, however. It is indeed ‘meaning’ — not blind algorithmic computation — that gives mathematics its substance. Fortunately, Gödel dealt formalism a devastating blow!”

So Penrose has little time for naysayers when it comes to mathematical (or, as we’ll see, essentially Platonic) truth. He makes that clear in the following passage:

“As to the very dogmatic Gödel-immune formalist who claims not even to recognize that there is such a thing as mathematical truth, I shall simply ignore him, since he apparently does not possess the truth-divining quality that the discussion is all about!”

[Note the word “divining”! Perhaps it was just a turn of phrase.]

Roger Penrose on Platonic Truth

The first thing to say is that if there is such a thing as mathematical truth (rather than mathematical correctness, formal consistency, etc.), then it must be very different from other kinds of truth. Take the following examples of the various theories of truth, if not truth itself: truth as correspondence, truth as coherence, truth as “the word of God”, truth as pragmatic utility, etc.

Roger Penrose himself acknowledges all this. Yet he still believes that mathematical truth is Platonic.

So what exactly does Penrose mean by the word “truth” in these contexts?

Penrose seems to assume that truth is part of (all?) mathematical systems when he says such things as “whatever formal system [a mathematician] might adopt”, that will “prov[e] his criterion of truth”. (On the surface, this statement sounds like a kind of formalism.) In other words, not much more is said here — or indeed elsewhere — about what mathematical truth actually is. Then again, Penrose’s Platonic truth may be so brute (or fundamental) that nothing much can be said about it.

So how does all this connect to Kurt Gödel?

Gödel’s Theorem, Algorithms and Consciousness

Roger Penrose explains what he takes to be the chasm which exists between the knowledge (as well as the use) of algorithms and the knowledge of mathematical truth. All this is primarily viewed in terms of Kurt Gödel’s first incompleteness theorem and its (to use Penrose’s words) “devastating blow” to formalism. Penrose’s position is expressed in the following passage:

“[T]he very essence of consciousness. It must be present whenever we directly perceive mathematical truth. When we convince ourselves of the validity of Gödel’s theorem we not only ‘see’ it, but by so doing we reveal the very non-algorithmic nature of the ‘seeing’ process itself.”

The words above state an explicit and essential link between consciousness and the ascertaining of mathematical truth. Basically, then, (human) consciousness is required to (using Penrose’s own word) “see” mathematical truth. This must also mean — to Penrose at least — that mathematical truths can’t be seen by computers or machines. However, that’s not because computerised robots or even rudimentary computers have no (non-biological) eyes, visual systems or intelligence (some would argue that they do) — but because they don’t instantiate consciousness.

Elsewhere, Penrose also wrote:

“[C]oncerning Gödel’s theorem, was that, at least in mathematics, conscious contemplation can sometimes enable one to ascertain the truth of a statement in a way that no algorithm could.”

Basically, the ascertaining of (Gödelian) truths is non-algorithmic. (Or, more particularly, ascertaining the truth of Gödel sentences is non-algorithmic.) Thus, algorithms can’t bring about (or simulate) the type of consciousness which is required to see Gödelian truths.

Indeed, not only is all this the case when it comes to Penrose’s position on consciousness generally, he even takes this position on what he calls “genuine intelligence” too.

Seeing Gödelian Truths

Penrose often uses the words “see”, “seen” and “visualised” when it comes to certain mathematical truths (as well as, so Penrose hints, other things — such as morality, “beauty”, aesthetical taste, etc.). That is, he believes that many mathematical truths are seen to be true without being proved to be true. So, in that simple sense at least, it can be argued that he’s simply putting Gödel’s own position (see the final section).

[Penrose repeatedly puts the word see in scare quotes . That is, he doesn’t mean literal seeing with the eyes.]

Along with “seen”, Penrose also uses the words “insight” and “intuition”.

For example he writes:

“[A] specific Gödel proposition — neither provable not disprovable using the axioms and rules of the formal system under consideration — is clearly seen, using our insights into the meanings of the operations in question, to be a true [ditto] proposition!”

Penrose isn’t the only one to use words like “see” in the context of mathematical truths.

For example, in the specific case of number theory and Gödel sentence G, the philosopher of logic Alasdair Urquhart uses the word “perception” (although it too is put in scare quotes) in the following:

“Since we do seem to have a ‘clear and distinct perception’ of the notion of truth in number theory, it has often been argued that this demonstrates a clear superiority of humans over machines.”

And, in the following paragraph, Urquhart continues:

“[We], standing outside the formal system, and using our mathematical insight, can see that the sentence G is true, and so we can surpass the capacity of any fixed machine.”

However, in the above it can be said that Urquhart is (at least in part) putting other people’s positions. And since I’ve just quoted Urquhart, it’s also interesting that he questions Penrose’s claim that he (or others) can see that a Gödel sentence is true. He writes:

“The problem with the Lucas/Penrose argument [] is that the key premise asserting that we can see the Gödel sentence to be true, remains undemonstrated. In fact, there are good reasons for thinking it to be false.”

In addition to the above, it also needs to be said that people may disagree as to exactly what it is they see. For example, one person may see (or intuit) that p is true, yet another person may see that the very same p is false. So even if we accept that there is Platonic seeing in both cases, that seeing alone doesn’t — and can’t — guarantee truth (or “truth without proof”).

Penrose also uses the word “sensing”. In this instance, Penrose goes beyond seeing the truth of a Gödel sentence and starts using much more modal and clearly Platonic ways of speaking. Indeed, he partly explains what he means by “seeing” here:

“[]I believe consciousness to be closely associated with the sensing of necessary truths — and thereby achieving a direct contact with Plato’s world of mathematical concepts.”

There is an important reason (at least within this specific context) as to why Penrose stresses Platonic sight. It’s because he believes that “sensing necessary truths [] is not an algorithmic procedure”. Indeed, this is part of Penrose’s wider stance against the possibility of (genuine) artificial intelligence.

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Part Two: Kurt Gödel Himself

When it comes to seeing the truth of Gödel sentences, intuition is said to enable (or bring about) such seeing.

The problem here is that in order to see the truth of a Gödel sentence (i.e., not simply recognise and understand what a Gödel sentence is), one needs to be proficient not only at mathematics, but also at metamathematics. So, basically, not many people can actually see the truth of any Gödel sentence.

Kurt Gödel himself believed intuition allows us to (to use his word) “grasp” various concepts. Like René Descartes’ position on what he called “clear and distinct ideas”, Gödel believed that if we grasp a concept (at least a concept of a specific type, though not always mathematical) in a clear way (whatever that may mean), then that will guarantee knowledge(?) of truth, as well as knowledge of (some kind of) content.

In detail, Gödel claimed that “we do have something like a perception also of the objects of set theory”. He also mentioned the “intuitive character” and “direct perceptibility” of mathematical objects. Yet, elsewhere, Gödel remarked that terms like “seeing” and “perceiving” (i.e., at least when it comes to sets, numbers, mathematical truths, etc.) mustn’t be taken literally. In his own words, Gödel argued that our “perception” of mathematical objects “must be totally different from [perceiving] sensual objects”.

However, it must also be said that Gödel’s admission (if that’s what it was) doesn’t work against the idea (or reality) of intuition. It’s simply an acknowledgement that the intuition of any given mathematical x isn’t identical to the seeing of, say, a red rose or to the perceiving of something’s being misshapen. In fact, Gödel believed that the kind of intuition he had in mind is actually more reliable that our perception of red roses (or at least judgments about them) or of other “external objects”. In other words, our (as it were) access to, say, basic mathematical truths, and even to the truth of a Gödel sentence, is more direct than our cognitive and perceptual access to a rose and its redness.

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Albert Einstein on Naive Science: Observation, Induction and Cataloguing

Albert Einstein once referred to the traditional view of science as one which posited a “continuous process of induction” and “the compilation of a classified catalogue”. He believed that this view “slurs over the important part played by intuition and deductive thought”.

In 1916, Albert Einstein offered his readers an account of what he believed traditional scientists — and most laypersons — took science to be.

Firstly, he mentioned the stress on induction. He wrote:

“From a systematic theoretical point of view, we may imagine the process of evolution of an empirical science to be a continuous process of induction.”

The basic take on science is that it is essentially inductive. Or at least this is what’s usually believed (i.e., by some scientists and philosophers) to be the contemporary layperson’s view. The problem here is that most laypersons don’t actually philosophise about science at all. More particularly, they rarely — if ever — use the word “induction” or “inductive”. That said, some philosophers tell us that a person may have the concept of a word without ever actually using the word itself. Thus, laypersons may have the concept [induction] without ever using the word “induction”.

What’s more likely is that traditional scientists — and laypersons — stressed observations. So Einstein continued:

“Theories are evolved and are expressed in short compass as statements of a large number of individual observations in the form of empirical laws, from which the general laws can be ascertained by comparison.”

Thus, “individual observations” are (or were) often believed to drive the entire scientific show.

[Once again, only empirical research can establish what most (or any) scientists and laypersons actually believe. Yet, anecdotally, this does seem to be the usual position.]

On this (as it were) naive picture of science, then, scientists simply go outside and … well, observe. Alternatively, scientists carry out experiments and then simply observe what happens.

Thus, according to Einstein’s take on the traditional take, scientists collect all their observations together into a large pot (or at the least they collect their “statements” about their observations together), and then they attempt to make sense of them. Or, in Einstein’s own words, scientists extract “empirical laws” (exclusively?) from their observations.

Clearly, innumerable other factors would be required in order to extract empirical laws from observations alone. And then the situation becomes even more complicated when “general laws” are “ascertained” from those observations and empirical laws.

So Einstein was correct to detect the naivety of this view of science. And that’s why he went on to write the following words:

“Regarded in this way, the development of a science bears some resemblance to the compilation of a classified catalogue. It is, as it were, a purely empirical enterprise.”

Thus, scientists are (or at least were) often seen as merely cataloguing nature (or cataloguing their observations of nature). This is almost like scientific (as it were) “stamp collecting”. (As is the case with some accounts of Francis Bacon and his own philosophy of science. See here.) If this process is followed, then, it was believed that everything could be kept scientifically kosher — or empirical. (The reader might have detected unwritten scare quotes around Einstein’s use of the word empirical.)

Of course, one can immediately ask why scientists were cataloguing the things they were in the first place. Why were they observing those parts of nature and not other parts? Why did they want to “compile[]” the things they compiled and not other things? In other words, there must have been prior factors — above and beyond what it is they observed — that brought about those very same observations.

Einstein himself then explained why this view is both simplistic and naive. He continued:

“But this point of view by no means embraces the whole of the actual process ; for it slurs over the important part played by intuition and deductive thought in the development of an exact science.”

It’s clear, however, that Einstein wasn’t actually entirely ruling out the traditional view of science. This meant (to Einstein) that observations — and even cataloguing — are indeed part of the story of science. That said, these things, according to Einstein, “by no means embrace[] the whole of the actual process”. And it’s here that Einstein adds “intuition and deductive thought [to] the development of an exact science”.

Einstein also wrote the following:

“The theory finds the justification for its existence in the fact that it correlates a large number of single observations [].”

That reference to a “correlat[ion] of a large number of single observations” is a perfect account of a particular kind of inductive process — enumerative induction.

For example, from the observation — and then correlation — of a large number of white swans, a subject may (or will ) conclude that “all swans are white”. Alternatively, a scientist may develop a (to use Einstein’s word) “theory” about swans and why they are all white. (This may even include natural laws of some kind.)

Einstein had also already mentioned “induction” (though in a critical way) when he wrote:

“we may imagine the process of evolution of an empirical science to be a continuous process of induction”.

One may now ask exactly how a scientist “correlates a large number of single observations”. (Alternatively: How does a scientist — or anyone else - connect the dots about all swans being white?) After all, if the theory “finds its justification [in the] fact that it correlates a large number of single observations”, one may suggest that theories were already required in order to enable those correlations. In simple terms, then, old and accepted theories would have been required (or needed) in order to find a new theory. In the white swans case, in order to conclude that all swans are white, the person who concluded that must have already accepted various other things about swans, the colour white, the whiteness of swans, the nature of observations, biology, ornithology, etc.

Einstein on Intuition and Deductive Thought

Earlier, Einstein was quoted stating that the naive view of science (i.e., discussed so far)

“slurs over the important part played by intuition and deductive thought in the development of an exact science”.

Einstein’s stress on what he calls “intuition” and “deductive thought” is a little odd. Many philosophers of science and scientists today would stress theory here — not intuition and deductive thought. Of course, theory may also be intimately tied to both intuition and deductive thought.

So, firstly, what about the word “intuition”?

In philosophy and mathematics, that word often has very specific and technical meanings (see here). So one wonders if Einstein used it in one of those technical ways himself. Perhaps, instead, Einstein simply meant speculation and/or theorising by the word “intuition”. That is, intuition (at least within a scientific context) is all thought which goes above and beyond the observational data. Indeed, intuition may also be required to make sense of the observations, and even lay the groundwork for observations.

So what about Einstein’s words “deductive thought”?

In a general and perhaps vague sense, if we have observations (or statements about them), then we can deduce things from those observations. That is, the observations don’t simply stand on their own. Scientists need to make sense of them. In addition, scientists can also deduce (not always logically) other (what Isaac Newton called) “conclusions” from them.

However, Einstein himself wrote that

“the investigator develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms”.

The quote directly above is Einstein (at least provisionally) treating physics as a kind of (pure) deductive logic. That is, instead of premises from which a conclusion can be derived (or axioms in mathematics which lead to theorems), we have observations and/or “fundamental assumptions” which lead to theories. And, in fact, Einstein himself says that “[w]e call such a system of thought a theory”.

Of course, much of what Einstein wrote about the traditional view of science is too neat and tidy. That is, scientific thinking and scientific practice didn’t really — or didn’t always — adhere to his retrospective formulations. But that’s often what happens in the philosophy of science.

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Max Tegmark Suggests the Term ‘Homo Sentiens’ to Account for Our Extra Specialness

The well-known physicist and AI expert Max Tegmark once wrote the following: “As we prepare to be humbled by ever smarter machines, I suggest that we rebrand ourselves as Homo sentiens!” Is this yet another attempt to place human beings at the top of the universal pile?

I came across a passage written by the Swedish-American cosmologist, physicist and machine learning researcher Max Tegmark in which he seems to be at pains to find abilities (or general characteristics) which truly distinguish human beings from all other… things in the Universe. Alternatively, Tegmark is simply expressing the general mindset of what he calls “we humans”.

This is the passage in full:

“Philosophers like to go Latin on this distinction, by contrasting sapience (the ability to think intelligently) with sentience (the ability to subjectively experience qualia). We humans have built our identity on being Homo sapiens, the smartest entities around. As we prepare to be humbled by ever smarter machines, I suggest that we rebrand ourselves as Homo sentiens!”

Human Beings are Extra Special and Unique?

Max Tegmark

So what’s the passage above all about?

Is it Max Tegmark’s expression of the strong need which so many human beings (or Homo sapiens) have to distinguish themselves from literally everything else?

For centuries, even for millennia, we humans have attempted to distinguish ourselves from other animals — and all else — in the Universe by our sapience. And now that this project has failed (as least according to Tegmark above), now it’s sentience’s turn to be our trump card.

So the same game simply repeats itself.

All sorts of things have been suggested to account for human specialness and uniqueness. That long list includes: language, opposable thumbs, walking upright, big brains, meaning, love, religion, empathy, the ability to play soccer, etc. Thus, the goalposts have kept on shifting. However, arguably some of these traits, abilities or characteristics are still deemed to make us human beings extra special and unique.

Yet, yes, it’s obviously the case that we human beings can do things which no other animal and no computer could ever do. No one would ever deny that.

Thus, human beings can create great — or even insignificant — works of art. No animal can do that. (Some computer programmes — arguably - can do the latter. See ‘Can Computers Create Art?’.) We humans can create religions, fly to the moon, cure diseases, build computers, “see” the truth of Gödel sentences, tell jokes, etc. No animal can do any of these things. And, as yet, no computer (at least without human input) can do any of these things either…

Actually, only a few human beings can do most — or even all — of the things just cited. However, perhaps all human beings have “the potential” to do all of them (see ‘Human potential’)…

But is that true? And what does “human potential” mean in this context?

So can we draw any general conclusions from these (seemingly) unique human abilities? And is there a single unique… something which characterises all of them and which is “purely human”?

As already stated, human beings are indeed unique. However, so too are cockroaches, ants, satellites, stars, bacteria, electrons, etc.

In addition, human beings can do things which animals can’t do. However, every animal can also do things which human beings can’t do.

So are there abilities or characteristics that we humans have, but which other animals and computers don’t have? And are these abilities more important and more unique than all the abilities which animals and computers have, but which we humans don’t have?

Or is it simply that we humans are unique in some extra-special way (or ways)?

Perhaps we are. However, how could that ever be established?

[Much of the above can be placed under the classification anthropocentrism. In addition, some responders have said that I’m really referring to Western culture — and its attitudes — in the above. Indeed, there may be an element of truth to that. That said, it depends on the non-Western cultures which do and don’t differ. And, in parallel, no blanket statement can be made about Western culture either when it comes to this specific subject.]

Sapience and Sentience

Max Tegmark himself believes that the meaning of the word sapience is “the ability to think intelligently”. Of course, the word “intelligence” is one of the slipperiest words around.

For a start, on many readings, an ant or even a cockroach is intelligent — or at least it acts or behaves intelligently. Indeed, the mathematical physicist Roger Penrose writes:

“[T]he behaviour pattern of an ant is enormously complex and subtle. Need we believe that their wonderfully effective control systems are unaided by whatever principle it is that give us our own qualities of understanding?”

And similar things can be said of most — or even all — computers (or computer programmes).

So we can’t decide if it’s (what Penrose — again — would call) “genuine intelligence” until we define “intelligence”. And that’s the problem! Thus, we’ve just moved from sapience to “the ability to think intelligently” to the word “intelligence” on its own.

Now what about Tegmark’s personal choice: sentience?

Sentience doesn’t really help us distinguish ourselves from a whole host of other animal species either. So, in that limited sense, surely sentience can’t play the role previously played by sapience (i.e., when it came to arguing for human specialness and uniqueness). Surely most animals must instantiate varying degrees of sentience too. And, as we’ve seen, perhaps that’s also true of sapience.

In neither the case of sapience nor sentience, then, do we have (as it were) ultra-uniqueness when it comes to human beings.

Finally, it can be noted that human(?) consciousness itself (i.e., not Homo sapiens generally) is said to fall into this category of extra specialness and uniqueness. (In a recent essay, I wrote on this subject.)

Consciousness or Human Consciousness?

In detail. It is argued that consciousness isn’t like other natural phenomena (such as photosynthesis, combustion, cognition and even life itself). Indeed, even many physicalists, naturalists, evolutionary theorists, neuroscientists, etc. freely admit that consciousness isn’t really like other natural phenomena. However, and in many respects, no given natural phenomenon is like any other natural phenomenon. (Think here of an electron’s charge, and then compare that to the mating habits of a baboon.)

Yet consciousness most certainly does have distinct features…

Yet so too does every other natural phenomenon. (Now think of how high a flea can jump relative to its size, or consider superfluidity.)

So are the unique characteristics of consciousness more unique than all these other examples of (as it were) natural uniqueness?

How on earth could a question like that be answered?

And isn’t it actually the case that we adult human beings take consciousness to be unique and extra special simply because consciousness is very important to us? In addition, isn’t all this at least partly down to the fact that we have (at least on most accounts) first-person access to our own consciousness?

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Note: Qualia

It’s interesting that Max Tegmark has used the technical term “qualia” in his definition of sentience. (The term is largely found in analytic philosophy, though it dates back to 1866.)

Even if we accept the existence of qualia (which will depend on what qualia are taken to be), do we actually “experience qualia” or is experience actually constituted by qualia? That is, is each experience (as it were) made up of qualia? Thus, surely we can’t firstly have an experience, and only then do we experience qualia. If that were the case, then an experience and qualia would be two separate things and/or two separate events (which would be separated in time).

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Isaac Newton’s Empiricist Philosophy of Science

Some critics (or downplayers) of the philosophy of science (or philosophy generally) may argue that Isaac Newton didn’t actually have a philosophy of science at all. Instead, he simply did what he did. That is, no matter how complex and reasoned Newton’s maths, observations, experiments, etc. were, it was still “just science”.

“Empiricism is the epistemology which has tried to make sense of the role of observation in the certification of scientific knowledge.” — Alex Rosenberg (see source here)

It’s true that Isaac Newton never used the words “philosophy of science” about his own words and work. Indeed, he wasn’t overly self-conscious about the philosophical and methodological underpinnings of his science. But none of this means that Newton didn’t actually have a philosophy of science or that he didn’t uphold philosophical positions (i.e., as they were directly relevant to his work) on the nature of science itself.

In addition, the classification empiricist science may seem like a truism or even a tautology. At least it’s what many laypeople (though not really many scientists) take science to be anyway. That is, science is often deemed to be mainly (or even only) about observations and experiments… Or at least many laypeople believed that it should be!

All that said, it’s now worth saying that these statements have nothing directly to do with the commonplace fact that science was classed as natural philosophy in the 17th century. In addition, it has nothing to do with the general idea that philosophical ideas and theories underpinned much science in those days (as they continue to do so today).

So, then, this essay is purely about Newton’s very own philosophy of science.

It’s also worth noting here (i.e., to set the scene) that David Hume’s more self-conscious and obvious general philosophy is in some ways like Isaac Newton’s philosophy of science. Indeed, often all one really needs to do is substitute Hume’s 18th-century term “impressions” (or “sense impressions”) for Newton’s earlier term “phenomena”. More relevantly, Hume believed that only concepts “derived from” impressions were relevant and significant. One can even read Hume as arguing that sense impressions are the exclusive source of all the knowledge we have of what he called “matters of fact”.

In terms of concrete examples, then, Hume had a problem with such “hypothetical entities'” as substance, vacuum, necessary connection, the self, etc. (All this can be found in Hume’s book An Enquiry Concerning Human Understanding.) And, similarly, Newton had a problem with such things as the aether, corpuscles and what he called “occult properties”.

All that said, an empiricist tradition can be said to go all the way back to Aristotle and even before him. After all, the ancient Greek philosopher did believe (or state) that “there is nothing in the intellect which was not first in the senses”.

Phenomena First

Isaac Newton stressed that scientists should observe what he called “phenomena”. (It’s hard to imagine many scientists — or even philosophers — not doing so.)

For example, Newton argued (as did Aristotle before him) that a scientist (or a natural philosopher) should carefully examine the world around him. In Newton’s own words:

“[A]lthough the arguing from Experiments and Observations by Induction be no Demonstration of general Conclusions, yet it is the best way of arguing which the Nature of Things admits of.”

It can be seen that Newton wasn’t a naive (or absolute) empiricist. In other words, he never believed it was literally all about “experiments and observations”. Yet, arguably, very few scientists — or even philosophers — have ever thought that way in any case. In Newton’s own example, then, he did say that he was arguing “from” experiments and observations. That is, he didn’t say that experiments and observations were the beginning and the end of all science. So Newton’s science was certainly no mere (as the phrase has it) “cataloguing of observations” either.

Still, Newton not being an absolute empiricist doesn’t mean that he wasn’t an empiricist at all.

Technically, Newton’s “conclusions” came from his experiments and observations. Thus, his conclusions weren’t merely (or exclusively) statements about the his actual experiments and observations. This also meant that any moves from experiments and observations to conclusions weren’t determinate or necessary. And that also had the consequence that Newton’s own experiments and observations (indeed all experiments and observations) could have led to different conclusions. Indeed, Newton was well aware of that.

Induction and Deduction

Newton believed that scientific processes must be kept in check in two ways:

(1) Inductive evidence must provide the groundwork of scientific work. 
(2) The consequences derived from inductive evidence must themselves be experimentally confirmed.

Of course, Newton’s use of the word “induction” needs to be fleshed out a little.

Arguably, there’s no such thing as a purely inductive process. And that’s even before any general “conclusions” are formulated.

Yet all this entirely depends on what Newton — as well as others — meant by the word “induction” in the 17th century.

More specifically, it’s a little difficult to know (or simply accept) what Newton meant by the phrase “arguing from Experiments and Observations by Induction”. That’s primarily because even during any inductive process there’ll still be other processes being employed. In addition, there’ll be theories, biases, prior knowledge, etc. which explain why those particular experiments were carried out in the first place. What’s more, it can be asked why Newton (or anyone else) observed those particular phenomena. Here again, theories, biases, preferences, scientific (as well as other) traditions, etc. must have been lurking in the background all along.

Of course, Newton did note that science involved both induction and deduction (i.e., not only one at the exclusion of the other). He did, after all, stress the drawing out of what he called “consequences”. Yet, here again, this almost seems obvious. Indeed, this stress on both induction and deduction can previously be found in the work of Roger Bacon, Robert Grosseteste, and, later, in the work of Galileo and Francis Bacon.

All that said, Newton certain did (as it were) come down on the side of induction (i.e., as against deduction).

For example, Newton once wrote that “particular propositions are inferred from the phenomena, and afterwards rendered general by induction”. This Newtonian and inductive methodology was applied to the following real cases. In Newton’s own words:

“Thus it was that the impenetrability, the mobility, and the impulsive forces of bodies, and the laws of motion and of gravitation, were discovered.”

But, here again, one can ask: Why these phenomena?

In this case, then, was Newton looking for the properties of impenetrability and impulsive force? And why did he assume any “laws” at all?

So it can even be argued that Newton’s own (as it were) grounding phenomena were already drenched in both theory and philosophy (or metaphysics) from the very beginning.

What’s more, even if Newton’s phenomena were pure, he still never discussed (let alone argued for) that transition from phenomena to the laws of motion or the laws of gravitation. As Rationalists may put it (to use a line of argument found in Laurence BonJour -see here), the phenomena may be as pure and empirical as you like. However, how did Newton (or anyone else) explain the links from the phenomena to any general laws or conclusions? After all, these links aren’t themselves phenomena and neither are they (i.e., in themselves) examples of induction. (This is the case even if the whole process itself can be deemed to be inductive in nature.)

Hypotheses Non Fingo

To put it simply: Newton believed that “theories” were acceptable, and that “hypotheses” were (largely) unacceptable. Of course, Newton used these terms in his very own (17th-century) way.

Yet, predictably, Newton broke his own rules on this strong distinction between theories and hypothesises (as we shall now see).

Newton’s term “theory” is used for that which can be “deduced from” what is observed and/or experimentally produced (or noted). Thus, inductive evidence can lead to a theory about such evidence. This Newtonian account of a scientific theory isn’t really a million miles from how the term is used today by educated laypersons and even by some scientists. However, Newton’s use of the word “hypothesis” is very odd to 21st-century ears.

For example, Newton once claim that hypotheses were (in relevant cases) about what he called “occult qualities”. And what are occult properties? In basic terms, they’re properties which can’t be observed or “measured”. (It’s odd, then, that some modern day — well — “occultists”, “spiritualists”, etc. often state “That’s just a theory!” to those scientists they disagree with.)

Thus, Newton didn’t like his own theories being classed as hypotheses.

A good example of Newton’s inductivist and/or empiricist inclinations (i.e., as they relate to this theory-hypothesis distinction) was his aversion to (using Stephen Toulmin’s phrase) “going beyond the phenomena”.

Take Newton’s theory of light.

Newton distinguished what can be observed from what may (or may not) underlie what it is scientists observe. In this case, then, scientists can observe certain properties of refraction. Thus, they can form a theory about such properties because they can observe them. However, they can’t observe what may (or may not) underlie such properties. In Newton’s case, then, scientists couldn’t observe (invisible) “waves” or “corpuscles”.

Now take the important — and similar — case of gravitational attraction.

Newton didn’t (or simply claimed not to) hypothesise about the underlying causes of gravitational attraction. Indeed, Newton famously stated the following three words: Hypotheses non fingo (“I frame no hypotheses”). Instead, and at least according to his own self-image, Newton simply noted phenomena and (as it were) observed what they did. And clearly — in this case at least — Newton couldn’t observe the underlying causes of gravitational attraction. (Some contemporary analytic philosophers may say that Newton couldn’t observe what they call “intrinsic properties”.)

The French philosopher René Descartes (1596–1650), on the other hand, did believe that he could hypothesise (or even know) about such underlying causes! He believed that gravitational attraction could be (or is) explained by vortices of aether.

Yet, despite all that, Newton did indeed hypothesise.

For example, Newton accepted that the aforementioned corpuscles and the aether may well exist. However, he still noted that scientists simply couldn’t observe such things. Thus, at best, Newton concluded that what came to be called “hypothetical entities” (or “theoretical entities”) may indeed be of some use in scientific research. However, scientists should never play fast and loose with such entities.

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