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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?

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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. Oct 9, 2023 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 9, 2023 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 Oct 9, 2023 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 9, 2023 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... Oct 9, 2023 at 18:39

7 Answers 7

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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”.

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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, 2023 at 6:19
  • 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, 2023 at 6:43
  • 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, 2023 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, 2023 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 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, 2023 at 9:50
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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.

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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, 2023 at 5:46
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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. Oct 10, 2023 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, 2023 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. Oct 10, 2023 at 11:30
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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. Oct 10, 2023 at 15:50
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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.

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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, 2023 at 8:49
  • "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, 2023 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, 2023 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. 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, 2023 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, 2023 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!

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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... Oct 11, 2023 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. ... Oct 12, 2023 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. Oct 12, 2023 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... Oct 12, 2023 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... Oct 12, 2023 at 20:57
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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.

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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?

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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?

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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, 2023 at 19:30
  • Sorry you don't see the need and while that's your choice, it's also a shame. Oct 10, 2023 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, 2023 at 19:41
  • Clearly, it's fine for you. Again, sorry you you didn't see the need. Oct 10, 2023 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, 2023 at 6:02

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