It looks like they chose to use the "universal gravitational constant" "k" instead of Newton^s constant, "G": p.23, "k^2 = universal gravitational constant, 1.32452139x10^20, m^3/(sec^2)(sun mass units)"
But the value and unit of "k" given in the Wikipedia page is different. Do you know what NASA document means by "universal gravitational constant" in modern sense?
The code appears to use in some places GK2M, which is the Newtonian constant of gravity, and in other places SQRDK, which is inappropriately described as a "gravitational constant", but it actually is the mass of the Sun expressed in some special units.
Newton's constant is known only with a very high uncertainty, i.e. a very low precision.
For the great bodies of the Solar System, e.g. the Sun and the planets, one knows with a high accuracy the product between their mass and Newton's constant, because that can be measured by the force with which they attract a body of known mass, e.g. an artificial satellite or an interplanetary probe.
Computing their mass in kilograms would be pointless in most cases, because that would introduce great uncertainties in the computations. So for the Sun and the planets one expresses their masses by the products between their mass and Newton's constant, whenever that is possible, i.e. whenever one needs to compute their attraction force exerted upon a small object.
Wikipedia names the product mass-Newtonian constant as "the standard gravitational parameter of a body", but I believe that this is a misleading name, because this product is just the mass of the body expressed in different (non-SI) units. Expressing a mass by its product with the Newtonian constant is not different from expressing the mass in pounds instead of kilograms. Using the Newtonian constant instead of some random unit conversion factor just has the advantage of removing the uncertainties from some expressions computing forces of gravity.
Gauss's constant k is defined as sqrt(G), but for a while the international standard was to define k and then compute G as k^2, which is why NASA refers to it that way.
I think it's just units. From wikipedia: "and its value in radians per day follows by setting Earth's semi-major axis (the astronomical unit, au) to unity, k:(rad/d) = (GM)0.5·au−1.5."
the value given in the paper assumes the distance in meters I think.
"If the axioms are true, and the subsequent reasoning is sound, then the conclusion is irrefutable. What we now have is a proof: something we can know for sure."
... if the axioms are true. We still don't know for sure absolutely.
"The idea of self-evident truths goes all the way back to Euclid’s “Elements” (ca. 300 B.C.), which depends on a handful of axioms—things that must be granted true at the outset, such as that one can draw a straight line between any two points on a plane."
Strictly speaking, Euclid does not state axioms. He starts with 23 Definitions, 5 Postulates and 5 Common Notions. Drawing a straight line from any point to any point is stated as Postulate 1.
Well, in modern mathematics we don't presume that the axioms are 'true' in any meaningful sense. All of modern mathematics is conditional.
So _if_ you find a system where the axioms of eg group theory hold, you can apply the findings of group theory. That doesn't make any statement about whether the axioms of group theory are 'true' in any absolute sense.
They hold well enough for eg the Rubik's cube, that you can use them there. But that's just a statement about a particular mental model we have of the Rubik's cube, and it only captures certain aspects of that toy, but not others. (Eg the model doesn't tell you what happens when you take the cube apart or hit it with a hammer or drop it from a height. It only tells you some properties of chaining together 'normal' moves.)
When you accept axioms randomly and reject even logical semantics, you wind up working with something like the so-called rado graph of Erdos. Call the first set of assumptions/structure meaningless and choose a new one, also having no grounding in reason - it turns out that would make no difference. "Almost certainly" you wind up with the same structure (the rado graph) in any case.
So you can reject meaning (identifying the axioms with simple labels 1, 2, 3, ...) and truth (interpreting logical relations arbitrarily), but in doing so you ironically restrict mathematics to the study of a single highly constrained system.
I only disagree somewhat though - it is all contingent. We do say something like
IF { group axioms } THEN { group theory }
and introduce even more contingencies in the application of theory. In a real problem domain these assumptions may not hold completely - but to that extent they are false, not having a meaningless truth value. Crucially, we then search for the right set of assumptions, and the conviction of modern science that there is a right set of assumptions is "effective" at least.
In the previous century there was an attempt to identify mathematics with formalism, but it defeated itself. We need to see truth in these systems - not individual axioms per se, but in systems composed from them. This is justified by the body of mathematics itself, to the non-formalist, the richness of which (by and large) comes from discoveries predating that movement.
I am accepting and rejecting axioms _arbitrarily_ depending on application, but not _randomly_, and definitely not i.i.d. random.
The Rado graph you mentioned does look very interesting as a mathematical object in its own right! But it has approximately nothing to do with what we discussed, exactly because it requires i.i.d. randomness.
> Crucially, we then search for the right set of assumptions, and the conviction of modern science that there is a right set of assumptions is "effective" at least.
Well, multiple sets of assumptions can be 'right' or right enough, or perhaps no set can be. Eg as far as we can tell, no clever sets of axioms will allow us to predict chaotic motion, or predict whether a member of a sufficiently complicated class of computer programs will halt. (However, we can still be clever in other ways, and change the type of questions we are asking, and eg look for the statistical behaviour of ensembles of trajectories etc.)
> In the previous century there was an attempt to identify mathematics with formalism, but it defeated itself.
I say it was extraordinarily successful! It actually managed to answer many of the questions it set out at the start. Of course, many of them were answered in the negative. But just because we can now proof that every formalism has its limits, doesn't mean that informal methods are automatically limitless. Or that there is any metaphysical 'truth'.
The 20th century saw enormous progress in terms of thinking about formalism abstractly (that's the 'self-defeat' you mention, thanks to giants like Gödel and Turing). But it's only in the last few years that we've seen progress in terms of actually formalising big chunks of math that people actually care about outside of the novelty of formalisation itself.
Btw, just to be clear: IMHO formalisation is just one of the tools that mathematicians have in their arsenal. I don't think it's somehow fundamental, especially not in practice.
Often when you develop a new field of mathematics, you investigate lots of interesting and connected problems; and only once you have a good feel for the lay of the land, do you then look for elegant theoretical foundations and definitions and axioms.
Formalisation typically comes last, not first.
You try and pick the formal structure that allows you to make the kind of conclusions you already have in mind and have investigated informally. If the sensible axioms you picked clash with a theorem you have already established, it's just as likely that you rework your axiom as that you rework your proof or theorem.
That bidirectional approach is sometimes a bit hard to see, because textbooks are usually written to start from the axioms and definitions. They present a sanitised view.
Rado has many interesting subgraphs, for example the implication graph of every countable theory. Are arithmetic propositions associated randomly? Not as we see it, but in some sense they could be (even iid). It is not the quite the same but I take the rado construction to be the logical extreme of the position there is no inherent/meaningful truth in such theories. You need to deal with this kind of many-worlds absurdity that falls out.
From your reply though I feel we are not really disagreeing so much. There is a kind of truth which is not propositional or self-evident but teleological. That is a sense in which I think the assigned truth values are meaningful.
The formalist movement was indispensable, I shouldn't have implied it was merely self-defeating. But I believe the philosophy that mathematics is fundamentally arbitrary mechanical symbol manipulation is wrong.
There is philosophy of mathematics. The below are my ideas of my philosophy of mathematics.
The axioms are true within the system that has those axioms. Therefore, the theorems that result from those axioms and rules, are also true in any system that has those axioms and rules.
It does not make it 'true' in an absolute sense (since it is 'true' within the system and any others (including the Platonic realism, and others too) that includes them), but absolute Truth is inexpressible (this is my conclusion from my study of mathematics and of philosophy of mathematics, but it applies to other stuff too).
However, you should avoid to be confused by such a thing, since some people apparently are. For example, just because some specific sequence of symbols has some use in some system, does not mean that it is the same in a different system (even if they can be mapped to them, which they often can be). Furthermore, even if "X OR NOT X" is true (regardless of what X is, as long as it is well-formed), that does not mean that either "X" must be true or "NOT X" must be true. And, just because values can be assigned to the symbols of classical (or other kind of) logic, does not make it necessary to assign those or any other values.
(Principia Discordia also has some things about "Psycho-Metaphysics".)
> Furthermore, even if "X OR NOT X" is true (regardless of what X is, as long as it is well-formed), that does not mean that either "X" must be true or "NOT X" must be true.
Independent of manipulating meaningless symbols, there's a whole branch of math called 'constructivism' where people try to find proofs without the 'law of the excluded middle'. Ideology / philosophy aside, the methods developed for this curiously handicapped game are of practical interest in computer science.
I am not sure this is true for "all" mathematics. You're still using some metalogical axioms that are always true. In particular, laws of untyped lambda calculus (or any other model of universal computation) are "axioms" that you consider unquestionably true (just like you can consider unquestionable that objective shared mathematical reality exists).
> I am not sure this is true for "all" mathematics. You're still using some metalogical axioms that are always true. In particular, laws of untyped lambda calculus (or any other model of universal computation) are "axioms" that you consider unquestionably true (just like you can consider unquestionable that objective shared mathematical reality exists).
Although, if you informally asked me if the rules of logic were true, then I would say that of course they are, if you asked me formally I think I would say that they are not unquestionably true, only unquestionable if you want to do classical mathematics. If you're willing to grant basic rules of logic, then certain consequences follow. If you're not, then you're not doing classical mathematics, although you might still be doing interesting mathematics—for example, if you decide not to accept the law of the excluded middle.
Yes, the law of the excluded middle is an interesting one to exclude. You could also try and go without certain infinities: eg you could remove induction (and things equivalent to induction) from your toolset, and see how far you can go.
There's quite a bit of ideology / philosophy about excluding the law of the excluded middle. But even if you set these aside, it turns out that 'constructive logic' without the 'excluded middle' has enormous practical applications in eg computer science.
True, and there is some variation in the axioms. But for the record, pretty much all systems keep the logical rules of AND elimination and OR introduction for example: if A and B are true, then A is true. If A is true, then A or B is true. However, the law of excluded middle is sometimes excluded for constructivist reasons.
Which axioms you take as true is a free choice. They aren't true or false by themselves.
What's irrefutably proven is that if you take this particular set of axioms, then these conclusions hold.
But you are free to choose other axioms, that will lead to other conclusions.
Some statements people use as axioms are equivalent (you can include one, and then derive the other and vice versa). Some are contradictory: you can include the axiom of choice or the axiom of determinacy, but not both as that will lead to a contradiction and thus an unsound system.
In a sense it's a matter of taste, mathematicians choose a set of axioms that leads to interesting things to think about.
What on earth do you mean by prove, If not within a system of proof?
You can prove the consistency of PBA, But you cannot prove that you cannot prove the inconsistency of PBA, because The inability to do both is the definition of consistency in the larger system.
"What's irrefutably proven is that if you take this particular set of axioms, then these conclusions hold."
This is what I tried to say in my comment. It's the author who talks about the truth of the axioms. I'm objecting to his claim that we end up with "something we can know for sure". No. Your truth depends on your assumptions.
As others have noted, axioms are more akin to a line in the sand—they are either so "obvious" as to be true or constitute such a useful and economic basis for further development that we decide to use them.
Mathematics is more about coordinating human observation and discussion than it is about capital T truth (unless you are a Platonist)
Even if you are a platonist, the conditional nature of axioms is still useful.
Eg look at group theory. It basically says, if you have a set of elements and some operation on this set that satisfies certain criteria (= the axioms of group theory), then you can draw all these conclusions.
I don't think anyone ever argued about whether the axioms of group theory are 'true' in the abstract, because everyone recognises that it depends on your application. Eg they are satisfied for the operations you can do on a Rubik's cube (especially a Platonic ideal of a Rubik's cube), but they aren't true for moves in Sokoban (even a Platonic ideal of Sokoban) nor Tetris.
More famously, look at Euclidean geometry: even setting aside curved spacetime of general relativity, even the ancients knew that Euclidean geometry isn't 'true' on the surface of a globe.
Axioms and postulates as Euclid uses that term are the same thing. In modern times we have gotten rid of the idea of statements that are self evidently true. So we don’t use the term postulate. We’d call Euclid’s postulates axioms today.
If we go down the skeptics' route, we can't know anything absolutely (except that we exist yada yada). But we still have to function in the real world, so we assume the most consistent observations will never change. From those observations, we extract the axioms, on which we build the tower of conclusions.
That talks more to how humans think than to what nature is. It is almost a philosophical debate but I feel gut instinct infinities are not really in nature.
Sure you can draw a good circle and say the ratio is Pi and that number encodes infinite information (albeit at high entropy) but to me that Pi is an algorithm for computing better and better approximations to a perfect circle, an object absent from nature.
I don't know QM but I suspect it is the same. It is our mental model. Using infinities is our concept. To me infinite things are algorithms (i.e. as X tends to infinity... Tends being an important word)
Pi very definitely does not encode infinite information. As you notice, we can write a very short computer program that will produce all the digits of pi (eventually).
In any case, you can do lots and lots of mathematics with either only finite objects, or if you allow limits as you suggest, you can do almost all of math.
Only a vanishingly small part of math deals with actual infinities in a way that cannot be re-written in terms of limits.
I thought it encoded all information but the catch is the index into it contains as much information as what you are trying to get out so it is effectively a very slow, negatively efficient zip format.
That's basically equivalent to saying that the natural numbers encode all information [0], it's just that the index to say which natural number you mean is rather long.
However for natural numbers, the 'decompression' is very fast. Much faster than for Pi.
That's right. But it does give a feel for what infinity is to our finite minds. It ain't just big numbers. It's all computation! (For a certain definition of computation)
Enumerating the natural numbers is only a single trivial computation. Enumerating the digits of Pi is a bit more involved, but it ain't universal, either.
Not a mathematician but my understanding is that the axioms aren't some universal truth that we discovered, but rather the foundation of our language of mathematics.
“the greatest scientist of all time is undoubtedly Sir Isaac Newton”
Newton routinely explained natural phenomena with supernatural causes, like God. In fact he overruled his mathematics with his supernatural explanations. A well-known example is his rationalization of chaotic behavior in his calculations of orbital motions with “the hand of God.” For this reason alone I would downgrade Newton’s scientific status and say that he was a prominent researcher in the 17th century but definitely not “undoubtably the greatest scientist of all time”. Today, we would not call someone who explains natural phenomena with supernatural causes a scientist.
It concerns the size of the largest sum-free set [0]. Take a (finite) set of integers, A. What is the largest subset of A such that no two entries sum to a third.
The previous results was not much better than |A|/3. The current, just proved, result shows that the largest subset is |A|/3 + c log(log(|A|)).
For example, the set {1,2,3} is not sum-free (1+2 = 3) but the subset {2,3} is sum-free (2+3 \notin {2,3}).
"It concerns the size of the largest sum-free set [0]. Take a (finite) set of integers, A. What is the largest subset of A such that no two entries sum to a third."
Yes, it seems to me we are focusing mainly about sets, not addition. Addition is secondary. Mainly I'm debating the title. The word "set" ought to be in the title too. I guess not a big deal.
Although as students we learn addition before we learn about sets, from the viewpoint of mathematics, sets are everywhere / everything can be expressed in terms of sets, so there's no point talking about sets in any given problem (unless it involves matters deep in set theory, which this does not). This is evident even colloquially, where we can talk about problems without explicitly using the word “set” — for example, “given some numbers, how many can you pick such that no two of them add up to another?” — while it's hard to avoid using “add” or “addition”.
So this problem is really more about addition than about sets, as the mathematicians who worked on it will say: the amount of set theory it involves is very little/almost nonexistent, while the properties of addition it involves are fairly deep.
(But sure, no harm if sets were mentioned in the title, I guess!)
This is a great question, and it's not as simple many comments here make it seem to be. This Veritasium video has a great explanation (and an enormous ball of molten sodium).
In case of Earth, Wikipedia describes [1] it as being "[..] generated by electric currents due to the motion of convection currents of a mixture of molten iron and nickel in Earth's outer core". This makes Earth a geodynamo [2]. (The aforementioned Wikipedia page is actually really long and detailed, a lot more than I would have thought)
