Ernst Mach's approach to physics definitions | Sociology and Pure Physics

[@markvincent8949]

Professor, the definitions for mass, force and density are complicated in another way. These quantities are idealised as a statistical aggregation. Mass is typically distributed over volume. The concept of a point mass like 'centre of mass' is effectively a black hole singularity. So if mass does not really exist at a point then is it a fluid of some kind at the smallest scale. Likewise a force is hard to pin down too. A force vector typically acting through a point would have infinite pressure or stress. A force is typically distributed over an area or volume but the physical form of force is a statistical mirage of convenience. Density is another nebulous quantity. Continuum fluid mechanics assumes fluid is a continuum but breaks down at small scales because the density fluctuates at the atomic level due to atomic motion. So solving the Navier Stokes equation based on continuum mechanical principles would actually give the wrong answer. Thanks for considering these conjectures.

[@njwildberger]

These are very good points Mark. Thanks for a clear enunciation of additional serious definitional challenges in physics.

[@ThePallidor]

Physics is notorious for using nouns when it should be using verbs. "Force" is a prime example. Or, "The field lines coming from the south pole of the magnet move the iron filings." It's a culture of equivocation much like modern math, but also a culture of reification where verbs are disguised as nouns to make it look like we're explaining something, like we're talking about underlying mechanisms rather than merely summarizing observations.

[@jimyocom8475]

Sociologists do that a lot, too. Though sociologists appear to be criticized for it a lot more.

[@dsm5d723]

Has anyone conceived of Navier-Stokes in terms of magnetic field lines? Seems obvious to me. But then again nobody understand what I say. I mean, force (magnetic field) distributed over volume is continuous between magnetic field lines, where symmetry breaks are more likely to occur. Or am I making no sense? Sort of the variable fluid flow of the Gaussian distribution. Is that worth a million bucks? The experts tend not to make sense to me. Norman is a glaring exception.

[@santerisatama5409]

What about manifold? Closest I can get to comprehending the concept is taxicab-norm which blurs out in distance into smooth curvatures. What I don't really understand is what we need the blurring for. I agree that how the question is formulated in the Millenium reward for Navier Stokes is too narrow and misleading.

[@peterjansen7929]

Thank you for another thought-provoking video. The problem you describe is probably common to most, possibly all, scientific disciplines. Ultimately they still rely on vague notions, an idea that an undefined 'something' is 'somehow' understood even though experts in the field never manage to clarify it properly. Usually, this is down to honest inability. Biologists struggle to define "life", and admit that they do. Worse things happen in mathematics. I had some maths lectures from a man, who started out with an open refusal to define a "set", merely requiring that in any given case it should be possible to decide unequivocally whether something was an element of that 'set'. The attitude towards honest inability to pin down 'knowledge' in most disciplines has been summarised very well by Emily Dickinson around 1863 in a kind of master-definition: "Nature is what we know - Yet have no art so say -" (From the poem «"Nature" is what we see -», No. 668 in Johnson's collection) Physicists unfortunately often resort to hidden circular 'reasoning' in attempts to 'explain' things. Most of us will know pictures of sheets of rubber deformed by heavy balls, with lighter balls 'orbiting' them for a while, 'explaining' gravitation with gravitation, never mentioning the obvious, that it isn't the shape of the sheet but the gravitational pull of the Earth that gives the balls their trajectories. That makes me go back to Dickinson: "During my education, It was announced to me That gravitation, stumbling, Fell from an apple tree!" (From «"Sit transit gloria mundi,"», No. 3 in Johnson's collection) Physicists also tend to advance from qualitative to quantitative descriptions, followed by a belief that a quantitative description only needs to be given a name containing the word "Law" to amount to an explanation. Thus, where there is a certain 'voltage' and a certain 'resistance', there is supposedly a certain 'current' BECAUSE OF "Ohm's Law", overlooking the fact that this 'law' allows us to calculate what will happen, to a sufficient degree of precision for an electrician's practical needs, but certainly doesn't explain anything.

[@Jeredin13]

This is so well thought out. Thanks for making this video

[@davidkeirsey9477]

Also another parallel definitional difficulty is in information science. The Framework of the 1+26 Finite Non-Abelian Simple Sporadic Groups, can be used to integrate the mathematics, physics, information science, and envolution (functional evolution) domains. Your decades of research and educational videos has served a part of this foundation. One can relationally associate and integrate these informational and mathematical concepts [relational information "atoms" and embedded maximal subgroups (information "molecules")] with informational and physical concepts of physics: Higgs Field (Monster Group), Higgs Boson (Janko 3 Group), Proton (Mathieu 11 Group), Neutron (Mathieu 12 Group), Strong Force (Held Group) are some of the example relationships. Keep up the good work.

[@siarez]

Thank you for this! Now I like Mach even more :) I look forward to your upcoming videos on these topics.

[@AnimeLover-su7jh]

Euler in a couple of books mentioned the fact you can not measure any quantity absolutely, it is only the measure of the proportion between two quantities of the same kind that we are measuring.

[@strangeWaters]

This video reminds me of a passage from Bishop's Constructive Analysis textbook: "A set is defined by describing exactly what must be done in order to construct an element of the set and what must be done in order to show that two elements are equal. There is no guarantee that the description will be understood; it may be that the author thinks he has described a set with sufficient clarity but a reader does not understand. ... it is impossible to consider every possible interpretation of our definition and say whether that is what we have in mind. There is always ambiguity, but it becomes less and less as the reader continues to read and discovers more and more of the author's intent, modifying his interpretations if necessary to fit the intentions of the author as they continue to unfold. At any stage of exposition the reader should be content if he can give a reasonable interpretation to account for everything the author has said. The expositor himself can never fully know all the possible ramifications of his definitions, and he is subject to the same necessity of modifying his interpretations, and sometimes his definitions as well, to conform to the dictates of experience." There is always ambiguity, but this isn't an excuse to flee into metaphysics. Formal systems need not rest on undefined entities -- just provisionally defined ones. Great video :)

[@ThePallidor]

In a single presentation, the key definitions need to be unambiguous to a degree suitable for the scope of the presentation. Or else there is no presentation, only a waffling. Definitions may need to be refined when the investigation is extended, sure, but in the extant investigation if they aren't clear there's nothing to present.

[@CandidDate]

@@ThePallidor Like all of "modern science." Definition #1: "Assume there is no God..." "All definitions to follow this (faulty) premise (or you won't get tenure)"

[@santerisatama5409]

I agree with Euclid and Mach. Good definitions are procedures of 'how to", constructive instructions. The notion of a "measurement unit" is not necessary, unless the need for such is specified and coherently justified. Some "definitions" are so fundamental and self-explanatory that adding a procedure to adds just confusion. Such is 'directed continuous movement'. The process of defining is a directed continuous movement. A computation is a directed continuous movement. By the constractive method of defining, can we define anything that does not involve movement? The hero of our story is known for saying: "Eppur si muove." What procedures by negation? Euclid's "Point has no part." and "Line has no width." may appear at first mysterious for a modern reader. Maybe the parsimony of those definitions is for didactic purposes, a DIY homework which mathematicians are still in the custom of giving? At the risk of spoiling, the intuitive meaning of those definitions was confirmed to me when I slow by slow realized the connection with the common notion "The whole is greater than the part.", by which Euclid defines mereology as an inequivalence relation, according to common intuition. Line having no width does not exclude that line can have depth. The definitions start to make sense when we realize that they are instructions for intuiting perspectives of projective decompositions. Intuiting dimensional decompositions and seeing the geometric perspective from 3D to OD with minds eye requires to imagine being flat-lander and line-dweller, learning the mathematical art of self-transformation into various perspectives, which then can be collected into a more holistic comprehension.

[@steveperkins511]

Hello Mr Wildberger. This is off topic, just taking the opportunity while there are few comments, but please please, if you possibly can, do look into a topic called 'Geometric Algebra', and it's extension, 'Geometric Calculus'. It is basically a new approach to vector algebra where perpendicular unit vectors can be multiplied to create 'unit bi-vectors', which are signed unit plane segments. These new 'bi-vector' objects have the property of squaring to negative one!! This occurs naturally and does not require us to create fictional entities with selected properties for our convenience. They are essentially the true 'imaginary unit' except they differ from 'i' in that they anti-commute. The perpendicular unit vectors e1 and e2 together with their product e1xe2 and the identity, form a canonical basis which is isomorphic to the dihedral group of 2x2 matrices you have worked on previously! You very nearly arrived at Geometric Algebra yourself! I am no professional mathematician, clearly, but can assure you that Geometric Algebra may well be the most important mathematics you have ever come across Professor Wildberger, it demands a re-write of all the text books! Basically, as I see it... "We forgot to multiply vectors!" , at least not fully anyway. The 'Geometric Product' as it is called goes like... "Geometric Product equals Inner Product plus Outer Product" where the inner product is the familiar scalar valued object and the outer product is a two dimensional planar object, something like the Grassman outer product (kind of) . The whole thing extends to n-d space and makes light work of 3-d rotations using tri-vectors, making conventional maths look clunky at best. Along with its extension called 'Geometric Calculus'', which gives us the equation... "Gradient equals Divergence plus Curl", it unifies many areas of mathematics and lends itself incredibly well to the physics of electromagnetism special relativity and much more! My terrible attempt an an explanation really doesn't do Geometric Algebra the justice it deserves but I highly recommend the channel 'Alan Macdonald' who has also written two books on the subject, for a basic intro. And another called 'BIvector', created by some computer scientists mostly, at the University of Utrecht, Holland, who have applied Projective Geometry to Geometric Algebra and taken classical physics further than any physicist or mathematician ever has, beautifully unifying the linear and rotational laws of motion using PGA (Projective Geometric Algebra). I have known of Geometric Algebra for seven years or more now and I cannot believe it has not yet gotten the attention it deserves. I really hope you will research Geometric Algebra and I look forward to hopefully watching a video in the future of you presenting your ideas. I can absolutely assure you that looking into Geometric Algebra will be a very fruitful use of your time! Finally, thank you Professor for all of your fantastic teaching. I have watched your channel for perhaps ten years or so and never commented before. Your work is both exceptional and inspiring! Best Wishes.

[@steveperkins511]

Sadly, they do all still use real numbers!

[@njwildberger]

Hi Steve, Of course this is a very good suggestion. I like to refer to "geometric algebra" as GCH algebra (which more directly gives credit to Grassmann, Clifford and Hestenes). This is indeed a very important topic, with huge potential for application in a wide variety of areas. Having said that, I do often feel uneasy when I read some treatments of this topic, especially when they use "real numbers" and "transcendental functions" like cos and sin etc. One of the exciting aspects of GCH algebra is the potential to bring truly rational thinking into a much wider range of areas, as there is a natural synergy between that topic and Rational Trigonometry and Universal Geometry, working over general fields and bilinear forms. It is one of my (long term) aims to study and layout this interesting area.

[@steveperkins511]

@@njwildberger Thank you for replying to me Norman. I am very happy to hear that this is one of your aims and I will certainly be looking forward to joining you for that.

[@CandidDate]

@@steveperkins511 But do they simplify understanding or calculations only with GA?

[@steveperkins511]

@@CandidDate yes most certainly

[@lazarbaruch]

As always, your presentations are extremely interesting. I want to point out that the root problem is the relation between force and mass, and it wasn't clear to me whether the second law isn't, in fact, not a law but simply a definition of the notion of force. I read carefully all that was available. Finally I came to a simple conclusion for the non-relavtivisc case: assume langth measurable by a stick, and mass by the quantity of matter, then we can define the momentum. We know from Galileo that if we change the speed, we make a change, but a change of speed is independent of the body in question. What else can we measure? Descartes observed that in the absence of external interaction, the momentum is preserved, so if we want to measure the changes in uniform motion, we can do this by measuring the changes in momentum. This way, we arrive immediately at Newton''s formulation of his second law. For the relativistic case, you can take momentum calculations in an interaction where one side is at low speed, and the other is at relativistic speed.

[@KarmaPeny]

As many others have said, thank you very much for another thought-provoking video. I was thinking about how maths concepts might be described in terms of 'explicit procedures' (as you suggest) when I encountered a maths forum comment that said "Do you consider 'f(n) = n + 1' a bijection between ℕ and ℕ+, where ℕ = {0,1,2,3,...} and ℕ+ = {1,2,3,...}?" In my view, the expression "f(n) = n + 1" denotes a programming function that takes a natural number as input and increments it by one. To me, it's simply a compact representation of a code snippet or algorithm translatable into various programming languages. While mathematicians may perceive it differently, perhaps as an infinite mapping between two infinite sets, I struggle to see it as such. Likewise, I don't view '√2' as a fixed value situated on an imaginary number line; instead, I regard it as representing a code snippet or algorithm that would perpetually continue if executed. While mathematicians may dispute whether a mathematical term like '√2' equates to a code segment housing a 'square root of two function', I anticipate they might acknowledge some form of connection between them. Essentially, I hope they would concede that one could be 'mapped' to the other. Now, let's contemplate a set of well-defined symbols capable of constructing functions related to real numbers. For instance, some symbols could define a 'square root of 2' algorithm, while others could depict a pi algorithm, and so forth. This task could be accomplished using a programming language or possibly by using existing mathematical symbols. Here's where it gets intriguing. With only a finite number of symbols ('x', say), there's a finite limit to the number of 'real number functions' achievable with 'x' symbols. Consequently, we can establish a one-to-one correspondence between each 'real number function' (formed using 'x' symbols) and natural numbers. As we increase 'x' to accommodate more 'real number functions', we can systematically continue to 'count' and thus map them to more natural numbers. Since 'x' grows infinitely (or without bound), no real number can elude our encoding into a function. Thus, for any conceivable specification of a real number, there will exist a mapping to an individual natural number. Hence, it seems we've uncovered a one-to-one correspondence between natural and real numbers.

[@santerisatama5409]

I think coherent mathematics can reject infinite loops by themselves as purely speculative, because they don't communicate anything with anyone. Recursive lazy evaluation (aka call-by-need) algorithms are another matter, as we get also something out from the process. If I understand correctly, lazy evaluation would work more efficiently on arbitrary word length architectures than these fixed word length architectures. The problem with the "real numbers" is not any the size the alphabet, but the fact that most real numbers can't be represented by any amount of characters of any size of alphabet. They simply don't have any closed form. The purpose of real numbers is to make something contious out of something discrete with maximal precision, which inevitably leads to point-reductionism and its denial of movement which denies computing itself ("zeno-machine"). What you are talking about is mapping a set of discrete symbols with another set of discrete symbols, and that doesn't satisfy the condition of continuum, without which the dream of real numbers does not serve any purpose. Nice try, though. A learning experience.

[@KarmaPeny]

​@@santerisatama5409Thank you for your considerate response to my suggestion. I fully concur with your observation that the issue with real numbers is that 'most of them' purportedly lack a closed form. Moreover, I would extend this point and query whether it is logical to assert their 'existence' if there is no means to unequivocally or precisely define them. However, I fear you may have misunderstood my argument. The scenario I propose bears a striking resemblance to the scenario outlined by the French mathematician Jules Richard, known as "Richard's paradox." Richard introduced his paradox in 1905, predating the existence of electronic computers or programming languages. Had these tools been available at the time, Richard might have employed them similarly to my proposal. In "Richard's paradox," he represents real numbers using English language descriptions, while I suggest code segments. He organises his descriptions by length and then alphabetically for strings of equal length. My proposed code segments, or 'Turing machines' if preferred, could be arranged in the same manner. For instance, consider the code segment for √2. It might encompass the code for a general square root function along with a line of code that could invoke that function, such as 'result=SQRT(number=2,base=10,significant_digits=0)'. These parameters would enable the function to execute for a specified number of significant digits (to correspond with the mathematical definition for a 'computable number') or to run unrestrictedly by setting the third parameter 'significant_digits' to zero. If, hypothetically, the most concise way to do this in a given programming language took up 300 characters (including the line of code containing the function call), then we could conclude that the code segment length corresponding to √2 is 300 characters long. Considering that there is a finite alphabet for any given programming language, there will be a finite number of code segments that are 300 characters long. Therefore, there must be a finite number of 'real numbers' that can be encoded (and thus considered well-defined or in a closed form) into a code segment length of 300 characters. Richard then defines another real number 'r' as follows: "The integer part of r is 0, the nth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of rn is 1." This 'r' value is akin to Cantor's anti-diagonal value, except that Richard is only applying it to definable/specifiable real numbers. And so with respect to Richard's paradox, we don’t need to consider whether or not any non-specifiable numbers can be said to exist. Richard argues that this is an English expression that unequivocally defines a real number 'r'. Thus, he presumes 'r' must be one of the rn numbers. However, he deems this paradoxical since 'r' was constructed to avoid being any of the rn numbers. In both cases, whether 'real number' is specified in English or via a piece of code, we describe an ongoing process. We start with shorter lengths and gradually increase to longer ones. Upon close examination, at any stage, we will have specified a finite number of 'real numbers' with self-contained code. However, the specification of Richard's 'r' number requires further consideration. Initially, I believed it would not be possible to specify a self-contained code segment to calculate Richard's 'r' number. However, upon further contemplation, I began to question whether I was correct in making that assertion. Its construction would necessitate a formulaic approach to creating the other number specifications. Then, conceivably, we could produce a single self-contained piece of code that would emulate the creation of the other numbers, calculate the nth digit of each of them, and output the altered digit as required. If such a thing were possible, then I would have to agree with Richard that his specification of 'r' could be described as unambiguously defined. However, as I delved deeper into this idea, it became somewhat mind-boggling to consider whether we could continue creating more of these 'r' values or anti-diagonals ('r1', 'r2', and so on). That is, could we proceed to create further code segments, each of which differs from all previous 'r' values as well as all specifiable numbers? Yet, while pondering this perplexing scenario, I stumbled upon a more fundamental issue that I had overlooked in my analysis of the original problem. If we assume that 'r' can be encoded into a piece of code, then what transpires during its processing when it has to deal with code segment lengths equal to its own? It appears that the code segment for 'r' would need to emulate or execute its own functionality and then apply further functionality to change the nth digit. This seems contradictory as it would require all of "its own functionality" plus "some more functionality" to be contained within "its own functionality," which is evidently impossible. Also it would need to represent not just its own real number, but its own real number with one digit altered, which is also impossible. Consequently, I reverted to my original belief that Richard's 'r' value is not well-defined as it cannot be constructed as a self-contained code segment. Richard concludes that his 'r' statement refers to the construction of an infinite set of real numbers, of which 'r' itself is a part, and so it does not meet the criteria of being unambiguously defined. Contemporary mathematicians agree that the definition of 'r' is invalid, but they claim it is because there is no well-defined notion of when an English phrase defines a real number. My proposed code segment approach would seem to negate this objection. Note that should mathematicians concur with Richard that the diagonal is not well-defined, it would suggest that Cantor's diagonal could not be defined, thus rendering the diagonal argument invalid. If all specifiable real numbers were said to already exist, all infinitely many of them, then Richard's description of 'r' would seem to be a valid specification (not only is the concept of infinite repetition readily accepted in formal definitions of real numbers, but the concept of Cantor's infinite anti-diagonal is also widely accepted by the mainstream). However, as such, the value it describes would need to already exist in the static set of all specifiable numbers. Hence, it would have to describe a value that is different from its own value, forming a trivial contradiction. Therefore, after much thought, I still maintain that the most reasonable resolution of Richard's paradox is that the concept of 'infinitely many' is incoherent. No other proposed solution can avoid contradiction to my mind (for the reasons explained above). It also renders all infinite diagonal arguments invalid.

[@santerisatama5409]

@@KarmaPeny Are you familiar with Bill Gosper's arithmetic of continued fractions? You might find Gosper's arguments for arbitrary word length machines instead of fixed word lenth machines very interesting. I've been musing that from at least from functional programming perspective, we can already consider blockchain programming a kind of arbitrary word length machine, the word being the blockchain. With this machine we can do mereological programming with Turing Complete blockchains, ie. generate and nest blockchains in blockchains. On the other hand, the big challenge of blockchain programming is the clocking problem, how to make distinct blockchains compute with each other in parallel manner (in simultaneous sync of same durations of clock ticks), without centralized "out-öf-sync middleman" intrachain translator. The blockchain clocking problem is mathematically analogical with the clocking problem of relativism, which is the central theme of Wildbergers approach in these lectures. Solve the intrachain clocking problem, solve relativism? *** I agree that Cantor's diagonal argument is nonsense. By nesting two distinct binary trees with each other in a kind of "entanglement", there's a mapping between the trees and the diagonal argument goes puff. Let me try to demonstrate, maybe you can make better sense of the issue than me. I'm a simpleton, so I start with alphabet of two marked characters, < and >. To define "something between" rationals, first let's construct rationals so that their "betweens" is easy to see. Let's concatenate mediants from the seed < >: < > < > < > etc. For numerical interpretation, < and > for numerator elements, for denominator element. Three distinct countables for multiset tally per word. Word < as a generator seed with corresponding numerical interpretations, and the combinatorics of combining < > and blanks into longer genesis strings with same condition of mirror symmetry, etc. Let's back to something more simple. The most basic structure that we've been focusing on so far, contains four distinct Fibonacci paths, listed here from L to R: A: etc. D: >>< etc. The distinct first two digits have the combinatorical form >. The A > correspond with Fibo paths where numerator > denominator (e-g- 8/5), and B and C >< with numerator < denominator (e.g. 5/8). The longest rational words by character count per row ar the B and C type Fibonacci rationals (their character count gives Fibonacci numbers; the character count of A and D type rationals gives Lucas numbers, ain't math beautiful?). Still leaving undecided the exact interpretation of the 1st digit of continued fraction paths, which side should be e.g. positive or negative - I like to leave that choice in quantum superposition - we can clearly see that the nth digit of the fibo path directly depends from the intial choises. Any case, because we have limited ourselves to binary alphabet, any and all choises are limited to binary swap. In other words, we have reduced Ricard's paradox about natural language to to De Saussure's structuralism of binary oppositions. What we can't avoid is that the oppositions are qualitatively distinct qualia, and we interprete them as such through holistic contextuality. Or quantum coherent quantum contextuality, if you like.

[@nikbl4k]

Norman i watch all your videos ever since i came across it from steve pattersons videos... great stuff.

[@Protoex]

Hi, professor. Are you familiar with the original work of ampere on eletrodynamics? Maxwell eqns even credits ampere to some extent, even though ampere had some fundamental disagreement to those ideas. A interesting aspect is the webber eletrodynamics as a theory of electricity that really satisfy the mach's principles.

[@yaroodevelop571]

Could You try to derive more pure physics formulas going from Special Relativity? (formula accoriding to E^2 = (mc^2)^2 + (pc)^2)

[@samuelarduino]

Critical Thinking!Prof. Wildberger do you have any comments on the work of Thomas Kuhn "The structure of scientific revolutions" in regards of this very fundamental topic? In my humble opinion Observing itself is allready a choise, judgement ,or an abstraction of reality ( maybe we observe the nonessential phenomena or the wrong wavelenght etc) Other topics to touch on in this respect are fundamental limitations neurophysiological of the human brain .(from prehistoric times,like patternrecognition, simplification, early opinions fight/flight, good/not good etc) the sociology of science tells it's own story. Highly appreciate your scientific and educational contributions on this forum!

[@peterrussell7846]

It seems to me that often discussions involving things such as square roots or trigonometric functions don’t actually benefit from using them. If they are used as a “handle” for an approximation for an approximate calculation then they appear to work but they have carried with them a gigantic amount of underlying “theory” about how you actually compute them, which isn’t necessary for the approximate engineering of physics calculation. On the other hand if you are using them to convey that you are actually computing something “exact” if you somehow assume you can ever get to their exactness then you haven’t really improved the situation, and a lot of the time you end up “reversing away” their presence. So for example you might just as well have conducted the discussion in terms of the “squared” quantity all along and not had intermediate steps involving the unattainable square root. So in a way I think this resonates with the idea that everyone has to be able to agree on what the procedure is for generating particular things and be able to write it down, because in such a context notions that involve gigantic amounts of baggage and doubt (like real numbers) are not adding anything useful - if it is an applied calculation they are unnecessary and if it is a pure calculation the level of befuddlement involved in conveying the idea is not making the calculation any more comfortably provable as being correct.

[@Yuri_Panbolsky]

"This discussion is particularly important when we try to understand Special Relativity." Help P. Marmet "Einstein's Theory of Relativity versus Classical Mechanics" This book demonstrates that using classical physics and Galilean coordinates, one can derive the observed phenomena attributed to relativity.Einstein's Relativity.. "The Collapse of the Lorentz Transformation".

[@christophergame7977]

According to a highly respected chap called Norman Wildberger, the ancient Babylonians used a strange and amazing kind of thing that is nowadays called a Pythagorean triple. They used such things to survey land. My great-grandfather was a surveyor. When I was a child, I learnt of a length measurement with 100 links in a chain. Procedure: For a primitive length measurement at terrestrial temperatures, choose a local effectively inertial condition, and put the chain between the two points whose separating distance you want to register and record, and count the number of links. If you want to measure the distance between two points that are much closer together than the boundaries of a block of land, then, again using an effectively inertial frame, put a well understood crystal between the two points, and count the number of atoms of the crystal. I think that atoms are more or less universal and relatively enduring existents. To me, this all smacks of natural numbers. Evidently, you won't actually get a universally exact distance measurement. You will have to be satisfied with integers. If you are interested in distances in a more general setting, such as in the interior of the sun, or between two stars, more elaborate schemes will be necessary. But I would say that the more elaborage methods should satisfy the general correspondence principle: under the original primitive conditions, the more elaborate distance measurements shall reduce to the more primitive measurements. For time, I still go for atomic clocks under the same primitive conditions. Again, I think that atomic clocks are more or less universal. A light clock has to have an arbitrary non-zero "distance" between its mirrors, is subject to variation in "length" if you believe in the Fitzgerald contraction, and must be oriented in a definite way with respect to motions of interest. I don't know how an atomic clock responds to direction or acceleration. I do not trust the celebrated and often postulated "clock hypothesis". Perhaps we don't need quantum mechanics itself right from the start, but it seems to me that we have to rely on the quantal nature of atoms as practically universal. The speed of light can perhaps be measured by use of measuring rods and clocks. As far as I currrently understand, the speed of light depends on gravity? I don't think it is appropriate to try to make our measurements depend on a coordinate system. Rather, I would say, let our coordinate system depend on our measurements. The leading principle here is that locality rules. We can be most confident of local measurements. Einstein supposed that the local conditions govern the local processes. I think he was right.

[@esorse]

Would it be fair to say that the reason for the convention of assuming there is a function called the "integrating factor", to solve a first order linear ordinary differential equation and a "successor function" that adds one to any natural number, along with induction in Peano's axioms, declaring that any natural number has a successor, is to create infinite flawless objects?

[@CandidDate]

Still waiting for the handheld electronic Rational Trigonometry calculator! Honestly, we have certain angles of arctan that can be expressed rationally, but not all for arcsin, arccos, etc., but you probably already know that these are useless due to their "real" qualities! Better to use Quadrance and Spread!

[@CandidDate]

Procedure for finding Square Roots: Say you wanted square root of c. set a - b*sqrt(c) then square this repeatedly. The square root will be approximated by a/b. Example 1 - 1*sqrt(2) squared equals 3 - 2*sqrt(2) . 3/2 = 1.5 close!. square again 17 - 12*sqrt(2) well 17/12 = 1.41666... closer! square again, 577 - 408*sqrt(2). well 577/408 equals 1.41421568... square again? ok, 665857 - 47032*sqrt(2) , again 665857/47032 = 1.414213562. Pretty darn close. Now square forever and you get an exact fraction! (it works better if "a" starts close to sqrt(c) and b = 1, but works for any c ! ) !!!

[@WK-5775]

Prof. Wildberger is not the kind of guy who will square anything "forever", I guess.

[@evanm2024]

With the "quibble" discussion, would you say that photons don't have momentum? Because it's something you define so that the "conservation of momentum" principle works, but it's not something that makes sense without this prior concept.

[@njwildberger]

The fact that photons seem to have momentum, despite the fact that the definitional formula for momentum implies that they obviously have zero momentum is just a good indication that the definitional formula (mass times velocity) of momentum is incorrect. Which it is, and that becomes apparent once we start to look at things relativistically, i.e. with the insight of Galileo's Invariance Principle.

[@ThePallidor]

"Photon" is never clearly defined either. In fact it's pretty explicitly equivocated on, even strategically.

[@mathewsamuel1386]

I believe the definition of physical properties of things should be theory independent. As such, definitions tied to measurements are not proper. That's why definitions of properties based on measurements tend to be unwieldy. We can see that in the different definitions of "mass" that it is not even obvious if mass is a property of matter anymore. When we make measurements, we do not actually measure physical ptoperties. Instead, we measure their effects on space and time. We should therefore define variables that describe these effects relative to how they are measured and then relate them to the theory independent definitions of the physical properties related to those effect capturing variables. For this reason, I really favor the definition of mass as the "quantity of matter constituting a physical object." Concepts like relativistic mass are no mass at all, but effects stemming from the behavior of material objects in response to forces. Quantum theory predicts the behavior of fields that can be described in terms of mediating particles, but since these particles are derived from the fields rather than having an independent existence of their own, I really think they should be described solely in terms of energy and no mass ascribed to them at all. I don’t think that the mass-energy equivalence principle is a good metaphysical warrant to express energy in terms of mass and vice versa. Physically, they are completely different things.

[@WK-5775]

Your belief is understandable, but it won't work, imo: for defining a physical quantity or property, you need some a priori understanding or picture of the phenomenon which you want to describe. For instance, can you separate the thing from the surrounding? Is it isolated in some sense, or is it something like a wave? A positive answer to the first question seems to be a prerequisite for assigning something like "mass" or "position" to it. And as to the surrounding: can it be described as some fixed "stage" on which events are happening? The answer to this decides whether you can work in Galilean or Newtonian setup, or in a setup of Special Relativity, of General Relativity, or of Quantum Gravity. (Or of something completely different, if you can think of it.) As I understand it, one important lesson to be learnt from 20th century physics is that one cannot understand what Nature "really" is; one can "only" (try to) describe and explain certain classes of phenomena.

[@mathewsamuel1386]

@@WK-5775 You can still make the definition a posteriori after you've understood the phenomenon enough to describe it in a theory.

[@WK-5775]

@@mathewsamuel1386 I totally agree with you. But this will definitely not make the definition theory independent.

[@mathewsamuel1386]

@@WK-5775 Why not? You mean you couldn't define mass as the quantity of matter constituting an object? Which theory determines that? I guess you may be confusing property with dimension, e.g., mass with the Kg, length with the meter, etc? That's not what I mean. Dimensions (units) can not be theory independent, but properties like mass, length, charge, etc., can all be defined in theory-free forms.

[@WK-5775]

@@mathewsamuel1386 Of course you can define the mass as "the quantity of matter" constituting the object under consideration. In classical mechanics it's a very useful notion, but it presupposes that you do have a distinguished object (and not a wave, for instance). In the easiest case, this object can be modelled as a point particle, i.e. without rotational degrees of freedom and without moment of inertia. Notice however, that the "amount of matter" definition relies in some sense on the picture that matter is infinitely divisible or at least that effects of binding energy (and the corresponding "loss" in mass, especially at the atomic or nuclear scale) is negligible. For me, such conceptual difficulties just show that the basic notions definitely are therory dependent. There is no way to build up a kind of "axiomatic" physics purely from observations. One always needs some hypotheses in order to interpret one's observations. (Primarily, I don't think of very complicated hypotheses involving high-level theory, but of very basic ones such as that a measuring device doesn't work differently when some experiment is repeated or when it has been moved to another place.)