# Do the limitations of science prevent the creation of only one true version of science?

I was recently intrigued by the following comment made by Ricky Gervais in this discussion with Stephen Colbert (Timestamp: 3:50).

If we took every science book, right, and every fact and destroyed them all, in a 1,000 years they'd all be back

I was curious as to the extent of truth in the above statement. It is obviously highly likely that notation will differ in these new books. My question, however, is directed towards the similarities in the underlying meaning of the old and new books. Considering that math and science, in general, is based on certain fundamental definitions, what is the likelihood of variations in definitions? More importantly, however, will different definitions create different versions of science?

To elucidate, I shall cite an example that illustrates present definitions on infinity. We may consider the infinite sum 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ... Fitting with our modern mathematical conception of infinity, the sum would be regarded as divergent. A new generation, however, may choose to allocate a value to the sum, using a technique such as Cesàro summation. The choice of attributing convergence or divergence to a series is a purely aesthetic choice. In the case of differing definitions of infinite sums created by the distinct generations, would this alter mathematical ideas?

In the above example, it seems to me that there is no correct answer. I want to also highlight a situation where I believe there is a correct answer but the correct answer is unknown. The video by YouTuber Veritasium Why No One Has Measured The Speed Of Light talks about the physical limitations of measuring the unidirectional speed of light. This has lead scientists to make the assumption that light travels at the same speed in all directions. My question is what if it was scientifically impossible to ever know an answer to the question "Does light travel at the same speed in all directions?". Wouldn't this obfuscate the possibility of the same assumptions being made by different generations?

To present some similar examples, analytic continuation may be an example of there being no correct answer. Further, Kurt Gödel's incompleteness theorems may have mathematical relevance with the second example.

I believe that the purpose of this post is to explore the inherent limitations of math and science. What do these limitations tell us about our reality if even the most sophisticated tool that is presently known to us (i.e. scientific reasoning) may be subjected to variations in definitions and may be characterised by an element of uncertainty?

My question may also have similarities with the following questions, which may help bring clarity to the question.

If we compared our scientific books with the books of an alien civilisation, to what extent would they be the same?

Is mathematics invented or discovered?

The question may also have relevance to this post (now deleted) on math.se.

• I don't think the mathematical definition of the sum of an infinite series is a simple matter of esthetics. I'm not very familiar with this area of math, but see the wiki pages on Taylor series and McClaurin series. Commented Apr 7, 2022 at 20:13
• @DavidGudeman As far as I'm aware, there is a choice of definition for the series as can be noted in the comment section of this answer. Nonetheless, I believe there must be instances of choice of definition in mathematics that influence the models built upon them.
– user58409
Commented Apr 7, 2022 at 21:43
• This kind of approach to science is totally useless, specifically in a philosophical context. Every reasonable knowledge of history and development of science supports the point of view that science produces limited and provisional knowledge. If this in unsatisfactory for someone searching "ultimate" truth, please leave science to itself and follow religion, magic, voodoo, etc: many options available. But, if so, please avoid using cars, cell phones, GPS, internet, weather forecast and every other type of "forms of life" based on SCIENTIFIC KNOWLEDGE. Commented Apr 8, 2022 at 9:58
• @MauroALLEGRANZA You write: If this in unsatisfactory for someone searching "ultimate" truth, please leave science to itself and follow religion, magic, voodoo, etc: many options available Isn't ultimate truth exactly what science looks for? Or is that to be found in astrology and voodoo? Commented Apr 10, 2022 at 23:56
• @Felicia Science is searching for an extension and ascertaining of knowledge claims, ie. improvement. Ultimate truth is where science ends as the practice does not make sense at that point anymore. Commented Apr 11, 2022 at 8:22

You really shouldn't ask so many questions at once, but to be fair many of them are connected. Indeed, almost everything I'm about to say can be summarised as follows: axioms are more or less useful in specific contexts which objectively exist in our experience, and trial and error gradually leads us to increasingly useful choices, which none of the concerns you mention really undermine. If they did, our civilization wouldn't have turned out the way it has. That's the tl;dr.

1. Can new generations create different axiomatic models?

In both mathematics and empirical sciences, we choose axioms whose consequences are useful for what we're working on. Borrowing a term from Daniel C Dennett's description of natural selection, we might call the best options, the ones we're naturally drawn to over time, "good tricks". The world is a certain way, and sooner or later practitioners of the scientific method would, through trial and error, find more and more useful axioms to understand that nature. Were humans not so capable, we wouldn't have seen scientific advancement on so many fronts in our world. (It's no good accusing me of circularity in arguing induction works because it has so far, because my reasoning is abductive: the simplest explanation of past successes is that nature is comprehensible - not that we can articulate what an incomprehensible world would be like, of course.) But this includes certain mathematics. For example, you can't help but find symmetries, and hence group theory, relevant across physics, and in particular in special relativity - which electromagnetism motivates - a generalization of groups called gyrogroups is a good trick.

1. If yes, can different axiomatic models create different models of (a) mathematics (b) science? If yes, is this a flaw of (a) mathematics (b) science?

On the one hand, yes, you can use different axioms. For (a), you can replace addition with Tsallis q-addition; for (b), you can pretend gravity obeys a different force-distance relationship. But we don't just coin axiomatic systems, we use them, and we find some pay off more than others in specific contexts. It's not a flaw; it's not a bug, it's a feature.

1. Does the unprovability of certain things such as the unidirectional speed of light leave fundamental holes in (a) mathematics (b) science?

We could discuss specific examples endlessly, but the crux is the most testable models are the ones that make the most tractable assumptions to address present concerns. In this example, why would you assume light travels in different speeds from A to B and from B to A, and similarly from C to D and from D to C, and yet no matter which way you point your apparatus the averaged bidirectional speed is the same for everyone? There's no theory that makes sense of why nature would be like that. By contrast, an isotropic speed of light is easily motivated by a theory that classical electromagnetism calls for, and which is empirically successful in other ways. In science, we respond to problems with ideas, and test those ideas against their other effects; an idea without other effects, such as light conspirising against us to seem like its speed is isotropic when it isn't, isn't good for anything, so it won't gain acceptance under such criteria.

Only true if we bother using the scientific method again fast enough, which an existing answer sociologically disputes, but that's not the point of your question. Gervais's real point is how this compares with, say, religions (each of which makes unique claims) and art/literature/music (which creates similarly unique works). There wouldn't be another Picasso without Picasso, but science is different because it uncovers how the world is - well, it probably approximately uncovers how the world is, before anyone gets pedantic and needlessly skeptical.

The choice of attributing convergence or divergence to a series is a purely aesthetic choice

Multiple definitions of the sum of an infinite series have interested mathematicians, but choosing one over another in a context is not about aesthetics. Mathematicians developed some theoretical insight into how these definitions can and can't work - in particular, which combinations of desirable properties can be combined, and which can't. Why are those ones desirable? Because they let us deduce the sum of one series from the sum of one or more others. But once all that's done, we can find certain contexts call for certain definitions. Physics loves one of them in particular, for this for example. The Casimir effect is measurably as strong as zeta function regularization lets us compute.

analytic continuation may be an example of there being no correct answer

A function can only be analytically continued in one way. I'm not sure whether your point is we might consider alternatives to analytic continuation, or might encounter functions with no analytic continuation. I think you already know my answer to the first; in case of the second, the functions without an AC tend not to unavoidably arise in situations where we need an AC. As a more general rule, functions that describe the world in scientific theories have the mathematical properties needed for the theory to be consistent and predictive, and the fact that many other functions don't have them isn't a problem; in fact, it helps us constrain what theory claims.

incompleteness theorems

Despite these theorems, a lot can be proven in specific theories; in our world, they often have been. If a cultural reset didn't lead to as much success, that may signify our descendants' bad luck, but not a very constraining fudnamental limitation on mathematics or empirical science.

What do these limitations tell us about our reality if even the most sophisticated tool that is presently known to us (i.e. scientific reasoning) may be subjected to variations in definitions and may be characterised by an element of uncertainty?

The incompleteness theorems aren't deduced from our reality, so tell us nothing specifically about it. It is true other axioms would have different results, but your focus has primarily been on mathematics rather than empirical sciences, whose methods make it easy to sniff out increasingly useful choices of axioms. But even in mathematics, such experimenting works. For example, in set theory we restricted comprehension for consistency, and added replacement, slightly more controversially added choice, and have ever since preferred to classify large cardinal axioms by their consequences rather than adopt them as a matter of concensus.

If we compared our scientific books with the books of an alien civilisation, to what extent would they be the same?

How advanced are they? If we visit them, then find them at a Neolithic level, they won't have such books. If they visit us, they understand enough physics for travel across vast distsnces. In that case, we are right about a lot, and they a right about a heck of a lot, and there will be much overlap. They won't, for example, disbelieve in electrons.

Is mathematics invented or discovered?

At the risk of controversy, I'll defend one specific answer: neither. Unger and Smolin 2015 combine two dichotomies: rigid vs non-rigid properties, and prior existence vs none. In their terminology, we discover that which meets the first option in each dichotomy, and invent that which meets the second in each. But mathematics is a third option they call evoked, meaning it lacks prior existence but has rigid properties once adopted. They also consider a fourth option they call fictional, which has prior existence but no rigid properties, such as biological species: they clearly exist, but out attempts to define them have to be somewhat malleable.

In Chapter 6 of Philosophy's Future: The Problem of Philosophical Progress, Pigliucci cites this four-category approach as an exampe of how one can make progress in philosophy by advancing a debate with an expanded taxonomy of ideas. It's also interesting for our present purposes because it allows us to summarize my position herein even more concisely as "we can do very well by evoking axiomatic systems to taste". Unger and Smolin also mention chess is evoked, and that provides an interesting analogy for our present purposes. You can invent endless alternative games, but humans are bound to come up with fun ones, so how our psychology defines "fun", unless also radically changed, constrains which mathematically possibe games we'll play. Admittedly, this is nowhere near enough of a constraint to imply a very chess-like game will emerge (I'll leave it to sociologists, historians etc. to say how simillar the most popular board games have been in different times and places). But physical reality is much more particular than the wide and personal range of things we find fun, which is why we'd expect a lot of science to be "unique" in practice.

To start, mathematics is not physics (which is related to science). Mathematics and logics are part of metaphysics, that is, to the realm of pure reason; which is not a posteriori from experience, but a priori. Mathematics is not science. Mathematics, in such sense, is discovered: it just expresses the mechanisms of thought. Reason determines the concepts that Mathematics express (see Kant's Transcendental Idealism). So, in a thousand years, if we survive, and don't evolve physiologically too much, the maths that describe our way of thinking (reason) will keep the same principles.

(Notice that the difference between discovered and invented is very subtle: math is a discovery of our mind, and it is an invention of our mind).

If by "likelihood of variations in definitions", you refer to changes in axioms or concepts (Google for formal systems and formal languages), axioms are probably not changing: math is just the formal language that expresses thought. Symbols might change, but the axioms,and perhaps the concepts will not.

So, speculating a bit, causality would still lead to the way of arithmetics; ideas will still lead to symbols and systems; judgments will still lead to relationships of ideas/systems and conclusions/theorems; induction and deduction will soon be identified as the meta elements of rules; understanding would raise kantian and aristotelian categories, which in turn will determine the need of symbols; aristotelian laws of though, which express the axioms of our thinking, will determine the rules of mathematics (the language), and some methodical and systematic work would allow growing, again, the body of mathematical knowledge (the tool). Arithmetics, with a bit of a posteriori physical knowledge, would allow geometry. A bit more of empirical rules (equivalent to the scientific method) would allow the basis for science... etc.

• "axioms are probably not changing". I do not see a logical justification for this claim. For example, the ZFC axioms are often debated. What prevents a new generation from altering these axioms?
– user58409
Commented Apr 9, 2022 at 11:43
• @CarefreeXplorer Nothing. But as said, maths express the way we think. If there are no changes in physiological evolution, the barber paradox would be confronted again, and a solution (equivalent to ZFC) will be needed. Commented Apr 9, 2022 at 11:56
• I don't see how same circumstances directly implies same outcomes. For example, same genes, same environment can lead to different personalities. I believe it is likely, in fact, that a new generation would encounter new problems, which would lead to new axiomatic models.
– user58409
Commented Apr 9, 2022 at 12:04
• @CarefreeXplorer you might be right, although I don't personally see the argument to accept that the axioms of thinking (the axioms of math) would change. A lot of mathematical discoveries were performed multiple times, by different people, in different contexts, with different semiotics, in unrelated environments and times (e.g. Ramanujan). Perhaps you should try an answer in such sense, I would be interested to read it. Commented Apr 9, 2022 at 12:56
• Perhaps it is inherently true of mathematics that even if there are axiomatic variations, the variations are limited in that they will lead to the same results. Or perhaps the axioms never change. But I can't find a satisfactory logical explanation for either.
– user58409
Commented Apr 9, 2022 at 13:10

Ricky Gervais claim is almost certainly false, for several reasons.

First, he assumes a type of sociological determinism, in which science is guaranteed to emerge from a rebuilt human society. However, we did not see science emerge from any human civilizations except for Europe during the renaissance. Sociological determinism relative to science is -- extremely implausible in the face of this contrary evidence.

Second, he asserts a time frame, 1000 years, that is extremely improbable. The development of science in our history took over 3000 years after the development of literacy.

IF we humans re-developed science, would it look the same? Maybe not. Per the Dunham/Quine principle, there are lots of alternative theories we could have for every science phenomenon: https://www.rit.edu/cla/philosophy/quine/underdetermination.html#:~:text=Underdetermination%20is%20a%20thesis%20explaining,face%20of%20any%20new%20evidence.

The same is true of math. Math is infinite. We could develop very different math.

The same is true of logic. Logic is plural https://www.cambridge.org/core/journals/think/article/abs/guide-to-logical-pluralism-for-nonlogicians/EDFDFA1C9EB65DB71848DABD6B12D877, and what logic we use is a user choice.

However, we humans have a biology, which includes predispositions, and we operate in an environment that has certain dominant features. these pragmatic considerations would incline future scientists to do science primarily reductively, think based on something similar to classical logic, and to focus on maths which match the dominant features of our world. For humans , the sciences, and math, would likely look fairly similar.

For aliens -- the predispositions would be different, and the environment may behave differently. Alien logic, maths and science, could well have very different foci, subject area divisions, and even include main categories of science that are minor in ours, and vice versa.

• The very first science that emerged was astronomy and that emerged in Babylonia. I know we Europeans like to think human civilisation began with us but that's not true ... Commented Apr 12, 2022 at 10:54
• @CarefreeXplorer -- I only answered your math questions indirectly, by noting that math is infinite. That there is a particular math that does X, we discover. We also discover what math applies to our world. The best model for this is Frege/Popper's 3 worlds ontology, which is summarized here: scribd.com/doc/7187000/Karl-Popper-Tanner-Lectures Commented Apr 12, 2022 at 15:01
• Also the options to match speed of light with more complex models, is just an application of the Dunham/Quine principle. I don't think the video is correct, I believe that directional red shift (there isn't one), and lasers operate through reinforcing of a frequency thru multiple reflections, and that is directional independent, and would not work with the 1/2c-infine option. But its PRINCIPLE is correct, one can almost certainly develop a more complex model that matches all our observations, per Dunham/Quine. Commented Apr 12, 2022 at 15:10
• @MoziburUllah -- Babylonian astrologers did careful observations, and postulated theories for how the stars controlled our fate. The concept of test/revision was less clearly implemented. The idea of test and revision on fate predictions has never been implemented in astrology -- instead confirmations are sought. Eventually, the problems with the dome/tent model vs observations. lead to the Ptolemaic model, and the beginnings of astronomy. But that wasn't the Babylonians. Commented Apr 12, 2022 at 15:45
• @Dcleve: I didn't mention astrology but both Kepler and Galileo were afficianados. We all know science relies on careful observations and such careful observations began with astronomy. In fact, its careful observations by astronomy that show the two great aporias in our current theories - dark energy and dark matter. The Babylonians got the Greeks started with observations ... science is not just constructing theories, it is also collecting data. Commented Apr 12, 2022 at 16:17

Mathematicians like sums. Since sums that diverge aren't useful, mathematicians have devised different criteria by which they can be evaluated. They are not aesthetic criteria but mathematical criteria.

Science before the word became appropriated by scientists meant simply an organised body of knowledge. Knowledge and knowhow has always been important (even neolithic man had to know how to make a stone axe). And as these generally accumulate, ways of organising them also become important. Science is often said to begin in modern times with the Renaissance. This is a modern myth and like most myths, untrue. What about building, astronomy, geometry and numbers? These are also sciences and they probably began in Africa.

If all the sciences died out and human knowledge had to begin again, then I expect the full range of human life and interests will be again be looked at but from a different angle, a different emphasis and obviously a different history.

We can look at ancient civilizations to help solve this problem. The Aztecs presumably developed their mathematical systems independently from the Arabs. The Aztecs had a base 20 system instead of base 10, complete with their own zero, fractions, addition, subtraction, multiplication, and division. Even though their mathematical system was different from ours, the fundamental arithmetical facts still applied, for example, 1+1=2 had an equivalent interpretation in their system.

Image courtesy of Library of Congress, Geography and Map Division

It may be possible that the Aztecs didn't know the Pythagorean theorem for right triangles. That is, the sum of the square of the sides is equal to the square of the hypotenuse. I think they probably did know this, since the 3-4-5 triangle (and other equivalents) are incredibly useful for agricultural layout. But if the Aztecs did know the Pythagorean theorem, then it would be the equivalent to that of the Arabs. It wouldn't fundamentally vary.

We could create a Venn Diagram of math the Aztecs and Arabs knew. The stuff in the intersection would be mathematically equivalent, even though the notation would differ. They would certainly have different estimates for values of pi, but they would know that a circle's circumference was pi times longer than its diameter.

Similarly, in Ricky Gervais' hypothetical situation, the language and notation would differ in the rewritten books. And that society would know some things ours doesn't and vice-versa. But, what we had in common would be equivalent. For example, their periodic table of the elements might be laid out differently, but would be fundamentally equivalent to ours, because the underlying chemistry and physics is unchanged. As another example, suppose instead of using bits (0 or 1) for computing, they used trits (0, 1, or 2). Some parts of their computer science would be vastly different than ours, but the shared fundamentals would be equivalent.

1. Can new generations create different axiomatic models?

Certainly, one could change Euclid's parallel postulate to get spherical geometry. This could theoretically occur for a sea-faring society.

1. If yes, can different axiomatic models create different models of (a) mathematics (b) science?

Yes, different math models can be created by changing axioms. This could also lead to different non-contradictory scientific facts, like great circle paths are optimal for navigating on a sphere, but Euclidean geometry works very well for agricultural layout.

1. If yes, is this a flaw of (a) mathematics (b) science?

No.

1. Does the unprovability of certain things such as the unidirectional speed of light leave fundamental holes in (a) mathematics (b) science?

We don't know a lot in both math and science. I don't think this is a problem.