Talk:Physics/Archive 6
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Newton quote 1
At the risk of recreating the dynamic of the development talk page, I present some annotation for Newton's quote below.
The definitions in Principia, which are not listed here, are part of the philosophy. But here philosophy refers to natural philosophy or what we refer to as 'physics'. The keyword 'principle' does not have a modern counterpart except for the term 'fundamental interaction'. These can be viewed as Newton's demarcation between what is physics, and what is mathematics.
- "I have presented principles of philosophy that are not, however, philosophical but strictly mathematical -- that is, those on which the study of philosophy can be based. These principles are the laws and conditions of motions and of forces, which especially relate to philosophy."[1] —Isaac Newton, Philosophiae Naturalis Principia Mathematica
Krea, it occurs to me that this quote might be data for the definition effort. Perhaps part of the difficulty in definition (which I suggest that we skate around right now) exists because we (our contemporary age) are too close to the problem. It may be easier if we just wait for more scholarship about the situation. But what will physics be like when we have gained enough perspective, enough distance from the problem of defining physics? It may be a long time from now. --Ancheta Wis 03:53, 21 September 2007 (UTC)
- Too close to the problem? Physics has been pretty well-defined for centuries, or at least generations. Here is Merriam-Webster's definition:
Pronunciation: 'fi-ziks Function: noun plural but singular or plural in construction Etymology: Latin physica, plural, natural science, from Greek physika, from neuter plural of physikos of nature, from physis growth, nature, from phyein to bring forth -- more at BE 1 : a science that deals with matter and energy and their interactions 2 a : the physical processes and phenomena of a particular system b : the physical properties and composition of something
- Here is Dictionary.com:
phys·ics /ˈfɪzɪks/ Pronunciation Key - Show Spelled Pronunciation[fiz-iks] Pronunciation Key - Show IPA Pronunciation –noun (used with a singular verb) the science that deals with matter, energy, motion, and force.
- That should pretty much settle the first paragraph of the lead. Let's move on. As for the boundaries between physics and philosophy (or physics and math, or physics and engineering, physics and chemistry, etc.), that subject is much too messy to deal with in this article for more than one short sentence. The bit from the previous lead about "physics emerged as a science distinct from natural philosophy" handled the issue quite well, and quite sufficiently. Gnixon 04:18, 21 September 2007 (UTC)
- A dictionary definition can be an inadequate characterization for some important topics. For example, the article truth has taken several years to get beyond the common answer; scores of editors have contributed to it. They started with the dictionary definition and proceeded to demolish it in the article on truth. Physics deserves no less. We don't have to start with a skimpy answer in the lead. It can be built up in stages and we have years to do it. Right now, we observe the spectacle of an definition akin to the four humors or Aristotle's four causes. But Newton's quote, coupled with his dynamical equations, is still applicable, even to relativity. So it's still right. To borrow a software analogy, the words of the dictionary definition are not strongly typed but are more akin to weakly typed or untyped. When matter and energy occur in the same series of words you know something is wrong. --Ancheta Wis 15:32, 23 September 2007 (UTC)
What's up with the History section?
Shouldn't the history of physics include more than Newton? Rracecarr 13:47, 21 September 2007 (UTC) Ok, the section has just been deleted by GNixon. Withdraw my comment. Rracecarr 15:02, 21 September 2007 (UTC)
- Since I don't seem to be alone here, I went ahead and cut it. Here is the excised content, for reference or rework. I recommend not trying to put in History until someone figures out at least a comprehensive, concise skeleton of a section. Gnixon 15:04, 21 September 2007 (UTC)
The emergence of physics as a science distinct from natural philosophy began with the scientific revolution of the 16th and 17th centuries, and continued through the dawn of modern physics in the early 20th century. Although physics is immense, it is still possible to glimpse it all from the influence of one physicist: Isaac Newton (1643-1727). Newton's work is an exemplar of a core theory of physics: mechanics.
- "I have presented principles of philosophy that are not, however, philosophical but strictly mathematical -- that is, those on which the study of philosophy can be based. These principles are the laws and conditions of motions and of forces, which especially relate to philosophy."[2] —Isaac Newton, Philosophiae Naturalis Principia Mathematica
From a few equations, Newton was able to posit a System of the World:
- "Seeing ... the variety of Motion which we find in the World is always decreasing, there is a necessity of conserving it and recruiting it by active Principles, such as are the case of Gravity, by which Planets and Comets keep their Motions in their Orbs, and Bodies acquire great motion in falling; and the cause of Fermentation, by which the Heart and Blood of Animals are kept in perpetual Motion and Heat; the inward parts of the Earth are constantly warm'd and in some places grow very hot; Bodies burn and shine, Mountains take fire, the Caverns of the Earth are blown up, and the Sun continues violently hot and lucid, and warms all things by his Light. For we meet with very little Motion in the World, besides what is owing to these active Principles"[3] —Isaac Newton, Opticks
But the science did not spring fully-formed from Newton. He had to discover some interrelations based on previous theory, experiment, and observation[4]:
- "In the beginning of the year 1665 I found the Method of approximating series[5] & the Rule for reducing any dignity of any Bionomial into such a series. The same year in May I found the method of Tangents of Gregory and Slusius, and in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions[6]. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force[7] with wch [a] globe revolving within a sphere presses the surface of the sphere) from Keplers rule of the period times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve; & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, and found them answer pretty nearly.[8] All this was in the two plague years of 1665-1666. For in those days I was in the prime of my age for invention and minded Mathematicks and Philosophy more then at any time since."[9] —Isaac Newton (as quoted in Richard Westfall's biography of Newton, whose mind was Never at Rest)
The subfields of physics were variously created and even unified during this time, as was the study of natural phenomena as positive sciences. Like Newton's reminiscence above, the interconnections between the various fields of study has been shown to be convincing evidence for the validity of science, taken as a whole. Thus sea-floor spreading serves as evidence for plate tectonics, and the cosmic microwave background serves as evidence for the big bang.
However, physics can lay claim to be the fundamental science, from which the others stem.
Old lead
I removed the in-line comments from the article, since they made editing quite hard. Here is a copy of the old version of the lead. I'm a fan of the 2nd paragraph (full disclosure: I wrote most of it), and I wonder if it could be incorporated in the current lead, which does a much better job of summarizing the article.
Begin quote:
Physics (Greek: φύσις (phúsis), "nature" and φυσικῆ (phusiké), "knowledge of nature") is the branch of science concerned with discovering and characterizing universal laws that govern matter, energy, space, and time. Discoveries in physics resonate throughout the natural sciences, and physics has been described as the "fundamental science" because other fields such as chemistry and biology investigate systems whose properties depend on the laws of physics.[10]
The emergence of physics as a science distinct from natural philosophy began with the scientific revolution of the 16th and 17th centuries, and continued through the dawn of modern physics in the early 20th century. The field has continued to expand, with a growing body of research leading to discoveries such as the Standard Model of fundamental particles and a detailed history of the universe, along with revolutionary new technologies like nuclear energy and semiconductors. Research today progresses on a vast array of topics, including high-temperature superconductivity, quantum computing, the search for the Higgs boson, and the attempt to develop a theory of quantum gravity. Grounded in observations and experiments and supported by deep, far-reaching theories, physics has made a multitude of contributions to science, technology, and philosophy.
End quote. Gnixon 15:58, 21 September 2007 (UTC)
The new lead also needs a paragraph about the core theories. Gnixon 15:59, 21 September 2007 (UTC)
- See comment below, for more considerations. --Ancheta Wis 17:28, 23 September 2007 (UTC)
The old lead is better. Defining physics as the science of matter gives the impression that physics is merely materials science. Definitions of physics given only in terms of matter (or matter and energy) do not provide a complete world picture of physics, as relativity also elucidates the structure of space and time. Energy, matter, space, and time are the fundamental constituents of the universe — physics is the science of the basic principles of the universe (Serway). --Kasparov 23:23, 27 September 2007 (UTC)
- Please give a book title, page number, and ISBN for the Serway quote.
- I like your characterization of 'fundamental constituents'. 'Basic principle' is reminiscent of Newton's quote 1 above, where he refers to 'mathematical principle' (i.e., equation). But what good is listing 4 items (Energy, matter, space, and time) when relativity shows 2 of the 4 are redundant? --Ancheta Wis 09:22, 28 September 2007 (UTC)
Here is some more material; I quote Sander Bais (2005) The Equations: icons of knowledge ISBN 0-674-01967-9 p.6 :
- "Mathematics as a language of nature ... These equations are compact statements about the way nature works, expressed in the language of mathematics. ... they have resulted from a critical dialogue between the observation of nature and the intuition and creative thinking of some great minds." --Sander Bais
--Ancheta Wis 09:38, 28 September 2007 (UTC)
Suggest we move section on thermodynamics and other bulk mechanics next to quantum mechanics
If we were to move the thermodynamics section below relativity then the transition to quantum mechanics would flow more naturally in the article. Blackbody discussions would then lead to Planck's discovery. Matter at low temperatures shows quantum mechanical effects at the macroscopic scale, as well.
As a side-effect, the relationship of Maxwell's equations to relativity would be more apparent, as an example of the covariance of Maxwell's equations under a Lorentz transformation.
So one diagram of interrelationships in the fields of physics would be
Mechanics +-> electromagnetism -->electrical circuitry -> electronics | | | +-> relativity (high speed physics) | +---> statistical mechanics -> thermodynamics -> mechanical engines | | | | | +-->low temperature physics --> low temperature quantum mechanics | +----->particle physics -->Big Bang (high temperature physics) | +<---->optics (note arrow goes both directions) --> astronomy | +----->Fluid mechanics -->aerodynamics, hydrodynamics, weather, plasma physics | +----->Materials science | +----->Photonics (electro-optics, optical computing elements, etc) | +----->Hall effect, etc
Note that it's naturally one subject when the subject is dealt with from its history. For example, Rabi's contribution to magnetic resonance would flow naturally to nuclear magnetic resonance, and to atomic clocks and timekeeping. From that, naturally to the GPS (Newton pops up again). From that, to the current work at NIST-JILA on Bose-Einstein condensates --Ancheta Wis 17:05, 23 September 2007 (UTC)
- Putting thermo between relativity and quantum wouldn't be unreasonable. The current order (mechanics, E&M, thermo, relativity, quantum) roughly follows the historical developments. I particularly like that the modern, 20th century theories of relativity and quantum are described together after the "classical" theories. Of course, thermo kind of transcends those classifications. Another possibility would be to put thermo last, since it draws from all the other theories. Gnixon 19:55, 24 September 2007 (UTC)
- The section has been moved down. Now to connect with Planck. --Ancheta Wis 10:35, 28 September 2007 (UTC)
Lead Section Comments
Right, a few things have been noted by several people these past few weeks, and so I'll try to address these as best I can. Right now, I'll only comment about the lead and leave the comments about the history section etc. until later. Also, in order to cut down on space, I'll not comment on points (about the lead) that I've agreed with and so you can generally assume this if I haven't mentioned it here.
About whether the new lead should have been put straight into the article, I'm only concerned with getting people talking and discussing the article and not so much with formality. Ancheta has made his opinions known about this, and mine follow those, so I won't repeat them again. But, I'll go along with whatever people decide should be done since I'm not really bothered about how we go about this, just that it gets done.
Concerning the definition itself, from the wip discussions, some editors stated a requirement for a "simple" definition of the subject. Gnixon has also said that the abstract definition of physics elucidated nothing practical about the subject itself. Conversely, Kasparov has now also mentioned the technical inaccuracy of the first line of the new paragraph. As a consequence of these consideration, I tried to include both in the new lead (which is why it contained statements like "in everyday terms": these were there to distinguish the different requirements for the definition). The dictionary definitions all give one way that you can define the subject and it is very useful for giving an impression of what the subject is, and it is also easily comprehensible in most cases. However, they are incorrect. Even the old lead was still technically incorrect. Kasparov, and many others, have mentioned that defining physics as the study of matter, energy, space and time was a good (technical) definition because it was the essence of the subject (since these were the fundamental constituents of that subject). This is not true: fundamental quantities also include things like colour. Also, and this is more pertinent, when we discuss physics at the ideological level, we must include things like the principle of relativity, and the principle of superposition and it would be hard to argue that these are any less important than energy, matter, space and time. Now, the point of all this is that we probably need a section all on its own to discuss these things (the quote that Ancheta gave from Newton was very interesting and I'd love to discuss it, but probably only in this new section). I am willing (and may in fact prefer this now) to go down this road and leave a simplistic definition in the lead section and mention that a proper discussion is made later. If we don't do this, however, I still strongly prefer to include both a simple and a technical definition in the lead. Whatever is decided please don't fault the simple definition for being inaccurate, or the technical one for being too abstract! But feel free to argue whether these should be included at all if you wish.
For the rest of the lead section, Gnixon has stated preference for, at least, parts of the old lead. Since I wrote most (if not all) of the new lead, I naturally prefer the new lead. But, I recognize this as my own personal bias, and, as I stated before, I prefer not to bicker about details. Thus I really don't mind which one (or more accurately, which parts of which lead) we go for. I'm just going to make two comments:
- Firstly, Gnixon and I both feel the need for mentioning the scientific method (which is important I think). The way that I went about this was to include some (very brief) context about physics' cultural and historical position with respect to the other sciences and civilization as a whole. I personally prefer to do it this way because it makes the section a bit more interesting quite frankly. But, again, I'm not too bothered about this, especially if anyone feels strongly about it.
- I deliberately didn't mention any specific theorems in the lead because quite often it ends up with people revering GM and/or the Quantum theory (and in some respects, it ends up like a popularity contest!). In reflection, perhaps we should mention something, but I would be careful about only including a theorem when it can serve to provide an example for a valid point being made, and endeavouring not to mislead the reader as to the theorem's importance with respect to other ideas in physics (i.e. not mention things like "physics is summed up by GR and the Quantum theory").
So, Gnixon (or anyone else), if you could modify the lead as you see fit, we can then make further comments about perceived strengths/weaknesses. Also, any opinions on whether we should dedicate a section to the definition alone would be welcome (I'm leaning for this to be so, right now); thank you. Krea 17:46, 1 October 2007 (UTC)
- Even if dictionaries are "technically incorrect," making that assessment is well beyond the scope of our jobs as Wikipedia editors. Good dictionaries are clearly WP:RS. Using such sources instead of trying to reinvent the wheel with WP:OR is an explicit Wikipedia policy. If dictionaries say physics is the science of matter and its motion, then that's what it is as far as Wikipedia is concerned. Let's just say so and move on. A clarifying sentence or two is fine as long as we don't go out of our way to dumb it down. The best way to explain to a reader what "physics" means is to give examples of what theories, research, topics, concepts, experiments, etc. fall within the subject. It's far too broad and fundamental to attempt an abstract discussion of definitions in the lead. On the other hand, I agree we can't afford to try and list all the big theories, concepts, etc. in the lead. Gnixon 20:04, 4 October 2007 (UTC)
- Is there original research? I thought it was fair to assume that definitions from textbooks would be a more appropriate source than dictionaries: a published definition from a physicist rather than an editor's interpretation of such definitions. Because of this, I don't think a dictionary's definitions are equal to these other ones. Is there a problem with this? Krea 03:32, 6 October 2007 (UTC)
Most compact statement of scientific knowledge
The Feynman lectures on physics I p.1-2 contains a provocative sentence with Feynman's opinion on the most compact statement of scientific knowledge. I appreciate that it was questioned in the revision history, so I included more of the quotation.
Feynman would give pronouncements like this to his students, and their clarity was alway stimulating; for example one of his lectures was on 'the method of solving any equation' which I have never seen in print. --Ancheta Wis 10:24, 4 October 2007 (UTC)
The flavor of his lecture style can be also seen in the first sentence of Feynman's lecture notes on The Theory of Fundamental Processes ISBN 0-8053-2507-7 p.1 'Review of the principles of quantum mechanics': "These lectures will cover all of physics." --Ancheta Wis 10:35, 4 October 2007 (UTC)
Energy absorption and emission in bulk matter
I propose that we smooth the transition from the statistical mechanics section to the quantum mechanics section with a few sentences about energy absorption in bulk matter: for example, the absorption and subsequent emission of energy in a block of bulk matter. This would unify the treatment of heat in a thermodynamic system, like a gas, with the behavior of a blackbody and the absorption and emission of light. This will lead to Wien's law, and Planck's correction, which leads straight to quantum mechanics. --Ancheta Wis 08:11, 10 October 2007 (UTC)
Fundamental definition
"Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge."
Couldn't physics be better described as the science of energy (which is the makeup of matter, motion, mass, charge and force (force can be considered a translation of energy)), space and time?
"As an experimental science, its goal is to understand the natural world.[6][7]"
Shouldn't we say "the universe", let alone the unnatural world as well? —Preceding unsigned comment added by 58.165.121.161 (talk) 09:24, 31 October 2007 (UTC)
- I personally have no problem with 'energy' but that lifts up the level of abstraction in the lead sentence, to something like "'energy' is the new water" of Empedocles' Classical elements, which have a checkered history. That is the reason that the lead sentence mentions 'matter and motion' in the same clause. The development article actually spent months on a lead, and you are welcome to look at it there.
- I personally have no problem with 'universe' but that would restrict the POV of the article to the cosmos and reduces the role of the microcosm, as in quantum mechanics.
- There is a problem with 'unnatural world', because it includes fantasy and other problematic statements of 'knowledge'. When you restrict yourself to statements which do not deal with untrammeled imagination, but which obey the strictures of scientific method, then at least you have a chance of stating a truth. Otherwise, there will be no difference between trash and treasure in the jewel box of natural laws which are commonly called the laws of physics.
- Your point is worthwhile and probably deserves a separate article on the difficulty of defining physics in a sentence or paragraph. My personal prejudice would be to list equations of physics, like Sander Bais does, in The Equations: icons of knowledge ISBN 0-674-01967-9. But that won't fly in the encyclopedia. --Ancheta Wis 11:00, 31 October 2007 (UTC)
- Note: World Cat has the ISBN entry wrong. Try Library of Congress instead, if you are following the ISBN --Ancheta Wis 14:53, 31 October 2007 (UTC)
Let me just say that I see your point. Firstly I can see how describing physics in one sentece by calling it the "science of energy" can lift the level of abstraction to many. I also appreciate that in existential circumstances such as those that you described, doing so can be confusing and perhaps even incorrect. I now better appreciate the merits of using matter and motion in the same clause. However I believe the first clause should and can define the topic entirely. Anything proceeding should only support this. I believe such a sentence would exceed the merits of the current one, the first clause of which only defines a narrow section of physics. If the fundamental makeup of the universe is energy, space and time, then our challenge here is the express physics as such, while contextualising in such a way as not to lift the level of abstraction.
Moving on, when we say "universe" we can contextualise in such a way that doesn't mislead some into thinking we are narrowing down to astrophysics. Further, don't get me wrong, I wasn't implying we should use the term "unnatural world".
On a side note, I agree entirely that physics is hard to define so concisely. Also, equations are just a way to express things. They can also be expressed with words.
A wild stab at trying to do what I've been talking about is here. If anyone has any other ideas, give it a shot.
"Physics is the study/science of energy, space and time. Energy is the makeup of all entities in the universe apart from space and time. Such entities include matter, mass, charge and motion. Force, another key entity studied in physics, can be considered a translation of energy. As an experimental science, the goal of physics is to understand the universe at all levels of scope." 58.169.193.41 09:21, 3 November 2007 (UTC)
- We appear to be on the same wavelength. Thank you. During the definition phase of the wip article, one problem which I encountered was which entities?.
- What your proposal appears to be saying is that as the fundamental science, physical laws apply to all entities; that as the earliest physicists observed the stars, and the motion of the planets, and the classical elements, some physicists were able to discover truths. Furthermore, by repeated application of these fundamental truths, the scope of physics has so far been found to apply to all entities, up to the bounds of space and time.
- In this way, the existence of the big bang was discovered, and our knowledge that time itself began in that epoch.
- In the same way, the knowledge that we ourselves (our physical bodies) came from the stars.
- In the same way, that energy is a common denominator by which entities with mass, charge, spin, etc. transform and translate.
- In this way, we see how, in the instants after the big bang, the fundamental forces separated and 'froze out' into the asymmetric forms we see today.
- Thus, the four rules of reasoning in Newton's program has succeeded, so far. By scientific method, which Newton sketched in the link, it is possible that some other phenomenon will be discovered which will overturn the brief world picture above.
- Sorry I have to leave right now, but the proposed definition sketched above appears to be consistent with physics up to General Relativity; the high-energy realm of the big bang (and the forms of matter at high energy) are ... .
- --Ancheta Wis 12:22, 3 November 2007 (UTC)
In answer to your first question: "All entities", well perhaps "physical entities" - otherwise the philosophers would trip us up on that one. By all physical entities I mean not just matter, but mass, charge, etc. I understand that entity is a generic concept, but isn't it fact that energy is the makeup of all physical entities? In response to your next passage, yes, this is an very interesting and eloquent way of saying that physics is based on energy and space-time. On a side note instead of "entities WITH mass, charge, spin, etc", why not "such as"? Your last comment interests me. If such a definition is not consistent with the development of General Relativity, then is the current one?...
Anon
- My problem with mass as an entity is that mass is an intensive attribute rather than extensive. That is like calling 2 an entity, when 2 is an attribute in my book. As you can see, we are right down to philosophy. But my answer might be one a physicist would give, saying that an integer is not real. So I guess it boils down to 'what is real' and for me, nature is real, true, observable by scientific method, repeatable by scientific method.
- When Newton defined mass points as his elements of computation, he derived the motion of the planets, You could argue that the mass points are entities. I agree that points with mass are entities, but that mass serves as adjective to point, which is an undefined term, according to my geometry teacher. So we are back to concepts which must remain undefined, just talked about, but used.
- If we were to expand on this, you might say that points were the basis for physical laws from the 1600s to 1900, and that they are the mathematical model for atoms. But when atoms were found to have structure, then we had to change our model to something that is more subtle. Thus the exploration of symmetry and strings in recent times.
- The nice thing about scientific method is that definition is not the starting point; instead, a characterization is the fundamental place to start. Characterization can include definition, but also observation and measurement, and information gathering.
- --Ancheta Wis 12:39, 13 November 2007 (UTC)
"American Heritage Dictionary - Cite This Source - Share This en·ti·ty (ěn'tĭ-tē) Pronunciation Key n. pl. en·ti·ties
1. Something that exists as a particular and discrete unit: Persons and corporations are equivalent entities under the law. 2. The fact of existence; being. 3. The existence of something considered apart from its properties."
"American Heritage Dictionary - Cite This Source - Share This point (point) Pronunciation Key n. ... Mathematics
1. A dimensionless geometric object having no properties except location.
... " I'm not sure how authoritative these are, however... Two things:
- 1. An entity having to be something "real". Now this is getting tricky, because how can we really define what is real, now that we're getting to the philosophical side of things. But lets boil it down even further to what "exists". Does something need to be (a)/(a group of) particles, em-waves, magnetic fields or what have you to "exist". The problem is that we could widen our definition of entities beyond nature that is "real", "true" and "observable", to other things. I think if we opened our minds up a bit, we could see how the spin/orbit of a nucleus/electron respectively (ie. mass, i hope) could be considered an entity, but, again, this is a matter of philosophy
- 2. Your taking of our idea of a point as an axoim. It is not an axoim (i hope) and any way of pretending it is so is a simplification (I would hope).
Interesting debate though... I guess it can't go any further. 124.177.66.44 (talk) 02:46, 29 January 2008 (UTC)
- ^ Isaac Newton (1687), The Principia: Mathematical Principles of Natural Philosophy, 3rd edition 1726. Translated by I.B. Cohen and Anne Whitman (1999, Introduction to Book 3, p.793)
- ^ Isaac Newton (1687), The Principia: Mathematical Principles of Natural Philosophy, 3rd edition 1726. Translated by I.B. Cohen and Anne Whitman (1999, Introduction to Book 3, p.793)
- ^ Isaac Newton (1718), Opticks, (2nd edition 1718, Book 3, Query 31, p.375)
- ^ See: scientific method.
- ^ See, for example how the Feynman Lectures on Physics Vol. I Ch. 22 Sec. 4, approximates a logarithm by linear interpolation using entries in a table of natural logarithms, to calculate irrational powers and the logarithms of irrational numbers. Note also Feynman's selection of "the most remarkable formula in mathematics" in Ch. 22.
- ^ See, for example how the Feynman Lectures on Physics Vol. I Ch. 9 Sec. 6 integrates Newton's dynamical equations from Sec. 5. This is a numerical approximation of what Newton referred to as his inverse method of fluxions. This method was the subject of bitter priority battles with Leibniz.
- ^ Galileo (1638), Two New Sciences mentions the 'force' required to move a ball up or down a ramp. See Hawking's section on Galileo, On the Shoulders of Giants, ISBN 0-7624-1348-4 pp. 524-525.
- ^ See Richard Westfall's biography of Newton, Never at Rest, for an account of Newton's construction of geometrical proofs, of the heuristic evidence he had found in 1665-6.
- ^ Richard Westfall, Never at Rest: A Biography of Isaac Newton (1980, p.143)
- ^ The Feynman Lectures on Physics Volume I, Chapter III. Feynman, Leighton and Sands. ISBN 0-201-02115-3 For the philosophical issues of whether other sciences can be "reduced" to physics, see reductionism and special sciences.