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Leibniz famously proposed a series of axioms or basic principles of nature -- or if one prefers Collingwood's terminology, a set of absolute presuppositions --, one of which was: "nature never makes jumps".

The phrase is sometimes expressed in Latin for good academic effect: natura non facit saltus, though Leibnitz originally wrote it in French (la nature ne fait jamais de sauts).

This principle implies that natural things and properties change gradually, in a continuous manner. So if you see e.g. a rabbit at place A at time T, and the same rabbit at place B in time T', then it follows that the rabbit has moved along a continuous line from A to B during the period T to T'. (Of course, rabbits do jump, but when they do so, their movement is still continuous)

My question is: is this principle still adhered to by modern science, or has QM contradicted it?

Olivier5
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    I would venture to say that quantum physics and general relativity don't fully decide the matter, even in rough concert. Research programs like causal set theory or causal-dynamical triangulation indicate a discrete world, string theory could go either way (the strings themselves being seemingly continuous internally?); some physicists propose QP as treating of "the absolutely small" but the neo-Kantianism of the Copenhagen interpretation seems to leave room for Kant's pro-continuity regulative principle, I suppose. – Kristian Berry Oct 09 '23 at 17:01
  • Thank you @KristianBerry, makes sense. Also thanks for the reference to Kant's formulation, which I was not aware of. – Olivier5 Oct 09 '23 at 17:29
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    He says that "we cannot make any determinate empirical use of this law, inasmuch as it does not present us with any criterion of affinity which could aid us in determining how far we ought to pursue the graduation of differences: it merely contains a general indication that it is our duty to seek for and, if possible, to discover them." link – Kristian Berry Oct 09 '23 at 17:34
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    @KristianBerry I.e. it's a metaphysical principle, as per Collingwood's definition of metaphysics having to do with absolute presuppositions. If this is the case, it cannot be proven or disproven by science. It can only inform science. – Olivier5 Oct 09 '23 at 18:06
  • Yet this inform is nothing but the spring of information arised from the unknown abysmal depths which might be very important for some obscure hard problems if not critical... – Double Knot Oct 09 '23 at 18:39
  • The "principle" predates Leibniz, it can be traced back to Aristotle, see Natura non facit saltus, who said that? Aristotle's context was rather restricted, and it provided some inspiration for evolutionary ideas of Lyell and Darwin. But generally, the "principle" is obviously false already in classical physics (there are plenty of discontinuous processes and forces) and was not "adhered to" even in Leibniz's time. – Conifold Oct 09 '23 at 19:34
  • The success of calculus doesn't even depend on nature not making jumps. Matter is quantized into atoms, so when you get down to the scale of an atom, nature makes a jump, but calculus still works on functions of mass, functions of the length of an object, etc. – David Gudeman Oct 09 '23 at 23:06
  • @Conifold Leibnitz formalized the principle as it is currently known, and that deserves mention. That some other dude might have thought of it before doesn't matter much. Please expand on classical physics making jumps. – Olivier5 Oct 10 '23 at 01:14
  • Even in classical physics there is a whole realm of phenomena that involve sudden and discontinuous changes. Catastrophe theory was developed to help study this. – Bumble Oct 10 '23 at 05:19
  • Interestingly, the "quantum" in "quantum mechanics" specifically refers to the existence of quanta of energy, meaning that the set of possible energy values for some things is discrete and finite, rather than continuous. – Stef Oct 10 '23 at 09:24
  • "The principle helped justify infinitesimal calculus, which Leibnitz also invented" : this claim is in error; see my answer. – Mikhail Katz Oct 10 '23 at 14:55
  • @KristianBerry In his Critique of pure Reason Kant speaks about a principle of continuity of forms: "a continuous transition from every species to every other species, by a gradual increase of difference" (B685). But he dismisses this principle, it is "a mere idea also because we cannot make any determinate empirical use of this law" (B685) Kant's statement seems outdated by today's method of gene sequencing the DNA of members of related species. – Jo Wehler Oct 10 '23 at 17:17
  • @JoWehler I should have said "proto-continuity," that would have conveyed my meaning better. I quote the same passage you secondarily quote in another comment above. – Kristian Berry Oct 10 '23 at 17:43
  • @KristianBerry You are right, I repeated the quote from your comment. My second reference should read (B689). - My comment wanted to emphasize that Kant's dismissal of the principle of continuity due to lack of empirical test is outdated. Do you agree? - What do you mean by "proto-continuity"? – Jo Wehler Oct 10 '23 at 18:44
  • @JoWehler Kant differentiated the range of (non)solutions to the size-of-the-understanding problem as finite-indefinite-infinite, with the median characteristic being the emblem of the physical world's partwise constitution, as neither purely discrete nor purely continuous, but an undecidable oscillation or fusion of these conditions. Or so it seems to me. For that, if my understanding of his understanding is correct, then the weak rational capacity to indefinitely divide the image of physical space/time, which he seems to admit, turns into a precursor to standard continuity, yet distinct. – Kristian Berry Oct 10 '23 at 22:47
  • As the question's author, I’m voting to close this question because I found the answers provided quite clear and useful, and I am concerned that additional debates might just muddle the issue. Thank you all for your contribution. – Olivier5 Oct 10 '23 at 19:35

7 Answers7

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Quantum mechanical wavefunctions are not quantized. They evolve smoothly (differentiably) in space and time. Measurable states are quantized.

Emergent phenomena produced by nature obviously can jump. Leibnitz would have known this, being familiar with reproduction, death and countable populations. (Populations increment and decrement; they do not smoothly vary.) So Leibnitz is just guilty of not magically knowing the right way to describe nature 200 years before anybody else did.

The statement,

"Given physically measurable observable X(t) at some point, if X(t=1) = A and X(t=2) = B and A<B, therefore exists some X(t) for some t:1<t<2 such that A<X<B"

is false.

The statement,

"Given a physical system whose physically measurable observables at some point may be stochastically predicted by a function Y(t), if Y(t=1) = A and Y(t=2) = B and A<B therefore exists some Y(t) for some t:1<t<2 such that A<Y<B"

is true.

g s
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    Death is a progressive process. A tree does not die in one second, but over several days. An animal, likewise, will progressively lose vital functions. The same applies to birth: embryology tells us how it happens progressively in animals, without any discontinuity along the way. In plants, we all know that plants produce flowers that produce seeds, without discontinuity. So your objection is incorrect. – Olivier5 Oct 10 '23 at 05:46
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    @Olivier5 Well, the objection does apply to things like radioactive decay and emission of photons. A neutron can decay and thereby produce a proton and an electron. According to our models, there is no time at which the electron is only partially created; it "jumps" directly from nonexistence to existence without any intermediate stages. However, the wavefunctions describing this process are continuous. One might say that the probability of the electron existing starts at 0 and then continuously increases. – Tanner Swett Oct 10 '23 at 11:18
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    @Tanner Swett Can you state the differential equation for the psi-function which supports your final claim? Why do you speak about wavefunctions(!) using the plural? – Jo Wehler Oct 10 '23 at 11:28
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    @JoWehler I only know tiny bits and pieces about quantum mechanics, so if any part of my comment seems inaccurate, it probably is inaccurate. – Tanner Swett Oct 10 '23 at 11:30
  • @TannerSwett the only thing I'd change in your statement is switching from Ultimate Real Reality related verbiage (created, existence, etc) to descriptions of measurement events: if we have a sample which we have just measured to be composed of n Cobalt-60 atoms, what is the probability that a future measurement will find a sample of n-1 Cobalt-60 atoms, 1 Nickel-60 atom, and a beta particle (a free electron)? – g s Oct 10 '23 at 14:43
  • @Olivier5 that isn't an argument against what I was saying, that is what I was saying. Death and reproduction are continuous processes. Number of trees increments and decrements. If you prefer counting seeds as fractional trees for some reason, we can use basketball scores, which increase in quantized units of 1, 2, or 3, even though basketball is a continuous process. – g s Oct 10 '23 at 14:48
  • @JoWehler I'm fine with plural wavefunctions. We need a different wavefunction to describe a free neutron, a 60Co atom, a 90Sr atom, etc. – g s Oct 10 '23 at 14:53
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    Another nice “purely physical” example of a discontinuous emergent quantity is something like “location of the highest point of earth’s surface”. Before the Himalayas formed, some other range — say the Scandinavian mountains, perhaps — were the highest. As the Scandinavian peaks eroded and the Himalayas were uplifted, at some point, the highest point “jumped” from a Scandinavian peak to a Himalayan. – Peter LeFanu Lumsdaine Oct 10 '23 at 15:50
  • @gs Surely, Leibnitz knew that one can count the number of people in a room using natural numbers... His principle still stand that nothing happens suddenly, at least in biology. – Olivier5 Oct 10 '23 at 16:00
  • @Olivier5 No, I disagree with you, Liebnitz did know that one can count the number of people in a room using natural numbers, and the principle that nature doesn't jump is correct! – g s Oct 10 '23 at 17:23
  • @JoWehler re: the equation which supports (a more measurement-oriented version of) Tanner's claim, you're looking for the transition moment integral. – g s Oct 10 '23 at 18:08
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    @PeterLeFanuLumsdaine you're essentially introducing non-local entanglement into classical physics, yet the frames of reference background-dependent laws of classical physics are local in nature without direct concern for most global physical properties of any rigid or elastic bodies such as your proposed highest point of earth's surface, except those conserved true quantities of motion such as energy or momentum of a closed subsystem within the whole world. Some completion theories of QM and GR are purely relational background-free then highest point of a whole may not make sense... – Double Knot Oct 10 '23 at 19:08
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The question whether “Nature makes jumps” cannot be answered: We do not have direct access to nature.

All that we know are the laws of nature. They are formulated by science. Hence they live on a theoretical ground, which is a different ontological domain than nature itself.

For a long time the principle “natura non facit saltus” served as a useful heuristics for science, guiding the search for the laws of nature. As noted in your question, the formal basis of these laws is the infinitesimal calculus due to Newton and Leibniz, in particular the concept of continuity.

Since the rise of quantum mechanics and its mathematical formulation using selfadjoint operators, the laws of physics obtained a discontinuous aspect, too: In general, the spectrum of a self-adjoint operator has besides its continuous part also a discrete part. Hence the interaction of two quantum mechanical systems may allow only a range of discrete measurements. In particular, the interaction of an atom with an experimental apparatus, i.e. of an observation, may produce measurements from only a discrete set of possible values.

We do not know how nature is when not observed. At best we know the laws when nature interacts with an experiment.

For more information about such questions from natural philosophy see e.g., “Copenhagen interpretation of quantum mechanics”.

Jo Wehler
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    Thanks, that makes sense. Then the question becomes: is the principle “natura non facit saltus” still today serving as a useful heuristics to guide the search for the laws of nature? – Olivier5 Oct 10 '23 at 06:19
  • As the formulation of quantum mechanics shows, the principle "natura non facit saltus" does no longer serve as a useful heuristics in the domain of microphysics. And note: Triggered by quantum mechanics the change of our conceptual thinking is even far more exceeding. – Jo Wehler Oct 10 '23 at 06:43
  • Thank you. I'm no qm specialist but it was indeed my hunch that the very concept of quanta seems at a variance with Leibnitz's principle of continuity. – Olivier5 Oct 10 '23 at 15:49
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    I'll tick your answer now. Just wanted to flag that @KristianBerry answered something very close in his comment, and deserves to be associated with the "tick". – Olivier5 Oct 10 '23 at 15:54
  • I would recommend that opinions about continuity vs discontinuity be stated carefully. These things belong to the domain of topology, and as demonstrated in https://en.wikipedia.org/wiki/Counterexamples_in_topology, our intuition is particularly fragile in that area. That said, I think in physics it is probably mostly a matter of what we actually mean when we ask the question; from some angles QM can be said to be not so much discrete in nature as "fuzzy". As for socalled quantum jumps, we don't actually know - we simply cannot observe closely enough to tell the difference. – j4nd3r53n Oct 12 '23 at 09:50
  • "... the laws of nature. They are formulated by science". I would prefer "approximated" or another non-deterministic word instead of "formulated". Science, while necessary and useful, is nothing more than humans' attempt to put natural phenomena into rules. This is evident in how new scientific discoveries disprove earlier scientific formulations. – Bram Vanroy Oct 12 '23 at 10:57
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“Nothing takes place suddenly, and it is one of my great principles that nature never makes leaps. I call this the Law of continuity.”

-Leibniz, in New Essays 56

And more generally the principle or postulate Natura non facit saltus, defended as a general principle of science, including by Darwin.

It really just comes down to defining jump.

The Chixulub asteroid provided a measure of discontinuity, that emptied many ecological niches and allowed rapid filling of them by animals more able to regulate their body temperature. We now understand Punctuated Equilibria to play the decisive role in evolutionary change like this, with short periods of intense selection and opportunity. This is very different to Darwin's picture, in which he defended this principle of continuous steady change.

The chromosome fusion of an ape that led to the hominid lineage, happened in a single individual once. In evolutionary terms, that is not a multi-generational change, it could be called a jump, & likely it came down to a specific unlikely event or set of events in a gamete or zygote. But that event itself was arguably continuous.

Schrödinger developed his wave mechanics in part to address the apparent discontinuity of 'quantum leaps'. We now understand electron orbitals as the Uncertainty Principle 'pushing back' on confinement of an electron around a nucleus, with the formation of standing-wave in it's probability distribution, and shifting to higher energy levels from the ground state as relating to the shift to a less confined and so higher energy standing wave, with consequent absorption of energy released on decay back as a photon. The real point there being, that even locality is continuous, there is no sharp transition from nothing to thing.

The implicit contrast being made by the principle, is with human actions. Like say CRISPR gene transfer, and instantaneous creation of a new trait. But CRISPR was invented by mimicking a natural process by viruses, albeit in nature with a much higher failure rate. And are human actions anyway 'outside' of nature? It feels like a somewhat arbitrary distinction to make.

We could identify as a distinctive quality of minds, to explore a large possibility space and then select one specific unlikely actual realisation, without enacting any or many of the irrelevant realisations. The general principle stands, that such behaviour has to arise out of non-mental circumstances, so the possibility of it must somehow be present or possible in inanimate matter. Panpsychism is one attempt to address this. I like David Krakauer's clearer term 'teleonomic matter' which he uses to describe what the domain of Complex Systems Theory is, that it's systems which get information about their environment, record it, and change dynamics in some way as a result.

Constructor Theory and Assembly Theory are attempts to quantify how this kind of 'sifting' of probability spaces can emerge.

Jens
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CriglCragl
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    Chixulub asteroid provided a measure of discontinuity - depends on what OP is actually meaning. If you look at the impact in incredibly high time resolution, then one could argue that it was a smooth process. Tracking from nanosecond to nanosecond there was a continual formation of the explosions, stuff going up in the athmosphere, individual plants and animals being pulverized (but still "slow" enough that you could consider them a "percentage of dead" over said nanoseconds) and so on and forth. Since OP has the qmtag, I assume that's what he's asking about. – AnoE Oct 10 '23 at 08:49
  • "The chromosome fusion of an ape that led to the hominid lineage, happened in a single individual once." If so, that individual would probably be unable to reproduce. It seems to me that we must postulate some reason making the change from 48 chromosomes to 46 advantageous or common (selection pressure, or a mutation) with the result that a generation of apes had at least one breeding pair of 46-chromosome apes. It may have taken millions of years for this coincidence to happen, which would either be a "jump" or not, depending on your definition. – Wastrel Oct 10 '23 at 14:33
  • @Wastrel It's even worse. If there's only one breeding pair in a location, they need to have several offspring and they have to interbreed to continue the lineage. – Barmar Oct 10 '23 at 15:22
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    @Wastrel Apparently it's not so uncommon. I found an article saying that fusion happens in about 1 in 1000 live births. https://www.kqed.org/quest/586/chromosome-fusion-chance-or-design. This is similar to the frequency of Downs Syndrome (an extra chromosome 21). – Barmar Oct 10 '23 at 15:27
  • @Wastrel And a Quora answer says that chromosome difference like this makes interbreeding harder, but not necessarily impossible, because when the fusion first happened the genes could still be paired up. – Barmar Oct 10 '23 at 15:32
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The original version of this question contained the following sentence:

"The principle helped justify infinitesimal calculus, which Leibnitz also invented."

There are some persistent misconceptions in Leibniz scholarship when passages in Leibnizian texts are taken out of context to justify incorrect claims. Take for example the following passage from New Essays:

In short, insensible •perceptions are as important to •psychology as insensible •corpuscles are to •natural science, and in each case it is unreasonable to reject them on the excuse that they are beyond the reach of our senses. Nothing takes place suddenly; one of my great and best confirmed maxims says that nature never makes leaps. I have called this maxim the Principle of Continuity. . . . This principle does a lot of work in natural science. It implies that any change from small to large or vice versa passes through something in between.

This is certainly an interesting and insightful passage. However, somebody who reads it without a preconceived notion that is related to infinitesimal calculus will notice that the passage talks about psychology and natural science, neither of which included mathematics (either in the 17th-18th century or today). Apparently the Principle of Continuity is unrelated to infinitesimal calculus.

There is a different idea in Leibniz called the Law of Continuity. One of its formulations is

the rules of the finite are found to succeed in the infinite and vice versa. (Leibniz to Varignon, 2 feb 1702).

As noted by Abraham Robinson, this is remarkably close to the transfer principle of infinitesimal analysis: if a formula holds for standard inputs, it will hold also for all inputs. For example, knowing that cos^2 x + sin^2 x =1 for all standard x, we would conclude that it holds for all x, including infinitesimal and infinite values. If anything, this represents a discontinuity: one jumps from finite to infinite values!

Mikhail Katz
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  • Your last quotation about law of continuity as noted by Robinson, is close to but not exactly the transfer principle which only applies to infinite objects R and star_R, say. Obviously card(x+1) = card(x) for infinite sets yet it doesn't hold for any everyday finite ones. Therefore this seems doesn't represent a discontinuity jumping from the finite to the infinite... – Double Knot Oct 11 '23 at 21:59
  • @DoubleKnot, the transfer principle applies to all formulas, and asserts that they are true for standard inputs if and only if they are true for all inputs. I am not sure what you mean when you say that it "only applies to infinite objects R and R, say". To elaborate on the example I gave in my answer*, if we have a proof of the formula cos^2 x + sin^2 x =1 for all standard (and therefore finite) inputs, the formula would be automatically true for all inputs, including infinite values of x. So I am not entirely sure what you are saying. ... – Mikhail Katz Oct 12 '23 at 12:27
  • ... I am open to the possibilit y that there may be a discrepancy you mentioned but would like to see further details. – Mikhail Katz Oct 12 '23 at 12:27
  • I mean the concept of internal (therefore finite) sets of R within the first order language of standard real analysis in Robinson's transfer principle needs to be emphasized, otherwise the famous Archimedean property would at first sight doesn't satisfy the transfer principle without introducing nonstandard hyperintegers in the sense of Skolem's nonstandard arithmetic model. In summary transfer principle is a purely internal principle when viewed from inside its applicable system, otherwise my above simple additive cardinality property for finite sets cannot transfer to its hyper system... – Double Knot Oct 12 '23 at 20:47
  • Said in another way: However, statements of the form "for any set of numbers S ..." may not carry over. The only properties that differ between the reals and the hyperreals are those that rely on quantification over sets, or other higher-level structures such as functions and relations...The transfer principle, however, does not mean that R and R have identical behavior...This is possible because the nonexistence of ω cannot be expressed as a first-order statement...* – Double Knot Oct 12 '23 at 20:57
  • @DoubleKnot, The situation is discussed in detail in my post here. In axiomatic approaches to infinitesimal analysis, the limitation you mentioned does not exist. – Mikhail Katz Oct 13 '23 at 11:57
  • Thanks for your (recent) post reference. I believe folks here are not familiar with Nelson’s IST axiomatic approach to NSA compared to the usual comprehensible nonstandard models from incompleteness theorems viewed internally or from outside. Btw it’s puzzling why Nelson believed and published inconsistency of PA at the end of his career given his IST background… – Double Knot Oct 14 '23 at 01:11
  • @DoubleKnot, Nelson was a powerful mathematician with many interests, including quantum theory, where he obtained significant results; see e.g., this publication. At any rate, returning to the classical model-theoretic approach to infinitesimal analysis (such as the approach pursued by Robinson), note that every classical entity (set, function, relation, ...) E has a nonstandard extension E, and the transfer principle applies. It turns out that one has an extra fringe of transfer applying also to things that are internal* – Mikhail Katz Oct 14 '23 at 17:52
  • ... in the R* world; this shouldn't be viewed as a weakness but rather as a strength. – Mikhail Katz Oct 14 '23 at 17:53
  • Thanks for you publication link though I cannot open it. Anyway I believe the seemingly 'surprised universal' transfer principle is just a justified 'reflection' of the reflection principle in set theory which essentially can reflect almost all 1st-order expressible properties except perhaps those relative properties such as countability/cardinality of the model as stressed by Skolem's paradox. Thus in this sense the discountinuity jump from finite to infinite values seems really about these relative properties... – Double Knot Oct 14 '23 at 20:04
  • @DoubleKnot, that's odd. You mean you can't reach the page https://u.math.biu.ac.il/~katzmik/infinitesimals.html or you can't reach any of the links at https://u.math.biu.ac.il/~katzmik/infinitesimals.html#15c ? – Mikhail Katz Oct 15 '23 at 11:47
  • No they keep loading and then saying 'This site can't be reached' – Double Knot Oct 15 '23 at 17:10
  • @DoubleKnot, That's odd. I assume the problem is temporary. If you are interested I can send you a reprint (but I would need an email address). Anyway, as far as reflection principle is concerned, it is indeed central to the proof of conservativity of IST over ZFC, but one doesn't really need it to understand the inner workings of infinitesimal analysis. – Mikhail Katz Oct 16 '23 at 16:30
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Philosophically speaking, and I saw a lot of appeal to the metaphysical presumptions of physicists, things have changed. That is, metaphysically, the notion of anti-realism challenges that there is a real universe that is either discrete or analog. The paradoxes that arise from quantum mechanics can be accounted for with an anti-realist epistemology (SEP) that says there is an external reality to those who observe it, but that reality isn't real in the sense that it's characterization exists independent of our observation and articulation of it. This was Kant's argument about the existence of the unknowable thing-in-itself.

I would say that while the realist metaphysical position is more popular among contemporary physicalists, popularity itself is not a criterion of correctness. If one abandons the premise that the universe is a dogmatic and real entity, and instead embraces a more process-oriented view that rejects disembodied objectivity for a social constructivist's notion of what 'universe' means, that is, a term around which a consensus is built on pragmatic considerations, then the question itself is rendered meaningless. Modern philosophy of science suggests that whether or not the decision to use a scientific theory with "gliding" real numbers or instead "jumping" integers to describe physical phenomena is at least to some extent underdetermined (SEP). Laws of Nature is an outmoded concept in the same way Laws of Thought is.

It is arguable that the defense of realist metaphysicalist positions is a function of the typical physicist's philosophical education (or lack thereof), and a longing for the certainty of philosophical positions before Kant to which Cartesian certainty inhered.

J D
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Yes, nature jumps. It does so whether or not we are watching. There's the old joke, maybe something on a t-shirt: "Jesus is coming, look busy! Nature is innately busy and dependably lawful. These are necessary ingredients of enduring dynamical structures. Raising her gaze from her navel to the world around her, a pedestrian observer would likely determine that the much of nature moves in a stepwise fashion, one steppingstone to the next, transitioning from one dynamic regimen to another. Would you say that a pendulum's moment of transition from peak potential to kinetic is continuous or discrete?

Don Foster
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Could you rephrase any of that? Do you want Leibniz' axioms or basic principles or Collingwood's absolute presuppositions… and which says 'nature never jumps'?

When you Post Latin, French or any other tongue don't you feel obliged to provide your idea of that in English; the more since your French and Latin don't seem the same? By QM do you mean quantum mechanics?

To me, for one, your moving rabbit analogy is about on the level of anything which needs an explanation like '… rabbits do jump, but their movement is still continuous…' which takes us back to the tortoise and hare. Remember them?

However much you conflate Leibniz and Collingwood, doesn't whether Nature makes jumps actually depend on how evolutionary or genetic mutation works?

From the point of view of an individual creature, any mutation is a huge change.

From the point of view of Nature or Evolution most mutations represent only a tiny step along the road of change.

Which would you like to discuss?

Robbie Goodwin
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  • Sorry, i don't see a need to rephrase. I'm grateful for the useful, interesting answers already provided, and am considering closing this thread down now. – Olivier5 Oct 10 '23 at 19:30
  • Sorry you don't see the need and while that's your choice, it's also a shame. – Robbie Goodwin Oct 10 '23 at 19:38
  • It's fine. Life is short and questions abound. Sorry if I came across as confused in the OP. – Olivier5 Oct 10 '23 at 19:41
  • Clearly, it's fine for you. Again, sorry you you didn't see the need. – Robbie Goodwin Oct 10 '23 at 20:00
  • Okay, for the record: 1) The reference to Collingwood was there because I'm a fan of his interpretation and analysis of metaphysics, as being about absolute presuppositions of this kind ("nature does not jump"). He did not comment on Leibnitz, to my recollection, and I was not trying to conflate the two, just to give a little 'advertisement' to Collingwood. His framing of metaphysics is illuminating and deserves to be better known and appreciated IMO. 2) QM: Quantum mechanics. – Olivier5 Oct 11 '23 at 06:02
  • I provided the English version. The Latin version was quoted because it is widely known, in spite of the fact that Leibnitz actually phrased his idea in French. Latin and French are not that hard to learn; otherwise I recommend DeepL, the best automatic translator out there. 4) The rabbit example was a reference to prestidigitation: when one sees a 'magician' pulling a rabbit out of his hat, one usually is amazed because that kind of discontinuity in the trajectory of rabbits is not supposed to happen as per the principle of continuity. I trust this clarifies.
    • – Olivier5 Oct 11 '23 at 06:02