Every die-hard fan of the scientific method knows that Karl Popper was a baller. While his achievements clearly extend far beyond analysis of the scientific method alone, he is arguably best known for his work on empirical falsification. In essence, the idea behind his argument is that a theory is only any good if there exists a direct and clear experimental/observational way to demonstrate that it is incorrect. In other words, it is more important to point out avenues in which your theory can be wrong than to flaunt all the possible ways it could be right.

Why am I writing about this? Mike and I just spent a week at 14,000ft on the Big Island directly searching for Planet Nine, and I’ve been thinking a lot about how Popper’s falsifiability criteria apply to the Planet Nine hypothesis… Obviously, if we search the entire sky at sufficient depth and don’t find Planet Nine, then we are plainly wrong. But I don’t think this is going to happen. Instead, I think we (or some other group) are going to detect Planet Nine on a timescale considerably shorter than a decade - maybe even this year if we/they get lucky. Which begs the question: if a planet beyond Neptune is found, how would we proceed to determine that the Planet Nine theory is actually right?

Figure 1. Mike and I at the telescope - where colors don't exist.

I’m sure this question sounds incredibly stupid, so let me back up a bit. The Batygin & Brown 2016 AJ paper is by no means the first to predict a trans-Neptunian planet with a semi-major axis of a few hundred astronomical units. That accolade goes to George Forbes, who in 1880 proposed a planet located at ~300AU, based upon an analysis of the clustering of the aphelion distances of periodic comets (sound familiar?). Since then, a trans-Neptunian planet has been re-proposed over and over again, which brings us to problem at hand: whose trans-Neptunian planet theory is right and whose is wrong?

In my view, there is a very clear and intelligible way to answer this question. Each proposition of a trans-Neptunian planet is uniquely defined by (i) the data it aims to explain and (ii) the dynamical mechanism that sculpts the observations. So in order to be deemed correct, the discovered planet must match both of these specifications of the theorized planet.

Figure 2. The current observational census of distant KBOs.

When it comes to the Planet Nine hypothesis, point (i) is well-established: Planet Nine is invoked to explain (1) physical clustering of distant Kuiper belt orbits, (2) the perihelion detachment of long-period KBOs such as Sedna and VP113, as well as (3) the origin of nearly-perpendicular orbits of centaurs in the solar system. Embarrassingly, until recently our understanding of the “machinery” behind how Planet Nine generates these observational signatures has been incomplete. That is, although we have plenty of numerical

*experiments*to demonstrate that Planet Nine can nicely reproduce the observed solar system, the*theory*that underlies these simulations has remained largely elusive.
The good news is that this is no longer a problem. In a recently accepted paper that I co-authored with Alessandro “Morby” Morbidelli, the theory of Planet Nine is characterized from semi-analytical grounds. So, for the first time, we not only know what Planet Nine does to the distant Kuiper belt, but we understand

*how*it does it.
The first lingering question that Morby and I tackled is that of stability: how do the distant Kuiper belt objects avoid being thrown out of the solar system by close encounters with Planet Nine, when their orbits intersect? Turns out, the answer lies in an orbital clockwork mechanism known as

*mean motion resonance*(MMR). When a Kuiper belt object is locked into an MMR with Planet Nine, it completes an integer number of orbits per (some other) integer number of orbits of Planet Nine. This strict rationality of the orbital periods allows the bodies to exchange orbital energy in a coherent fashion, and ultimately avoid collisions.
But how do such configurations arise in nature? Remarkably, the answer in this case is “by chance.” When the Kuiper belt first formed, a staggering number (roughly 30 Earth masses worth) of small, icy asteroid-like bodies were thrown out into the distant realm of the solar system by Neptune (for the interested reader, see papers about the Nice model here and here). Most of these objects were not fortunate enough to accidentally land into mean motion resonances with Planet Nine and were ejected from the solar system. However, the few that were, survive in the distant Kuiper belt to this day, and comprise the anti-aligned cluster of orbits that we observe. As a demonstration of this point, check out the simulated orbital period distribution of surviving Kuiper belt objects in one of our idealized simulations, and note that all distant bodies have rational orbital periods with that of Planet Nine:

All of this said, the full picture is of course not as clear-cut. Within the context of our most realistic calculations of distant Kuiper belt evolution, the clustered KBOs chaotically hop between resonances, instead of staying put. Still, the qualitative framework provided by analysis of isolated resonances holds well, even in our most computationally expensive simulations.

Ok so this resolves the question of how Kuiper belt objects survive, but it leaves open the question of why their orbits are clustered together. Intriguingly, a qualitatively different dynamical mechanism - known as

*secular*interactions (see here for a neat discussion) - is responsible for the orbital confinement that we see. Plainly speaking, over exceedingly long periods of time (e.g. hundreds of orbits), Planet Nine and the Kuiper belt objects it perturbs will see each-other in almost every possible configuration along their respective orbits. Thus, their long-term evolution behaves as if the mass of Planet Nine has been smeared over its orbital trajectory, and its gravitational field torques the elliptical orbit of the test particle. The plot below shows the eccentricity-longitude of perihelion portrait of this secular dynamic inside the 3:2 mean motion resonance, where the background color scale and contours have been computed analytically and the orange curve represents a trajectory drawn from a numerical simulation.
Indeed, the fact that the semi-analytic theory predicts looped trajectories that cluster around a P9 longitude of perihelion offset of 180 degrees implies that the raising of perihelion distances (i.e. lowering of eccentricities) of long-period KBOs and anti-aligned orbital confinement are actually the same dynamical effect. In other words, the reason that objects such as Sedna and VP113 have orbits that are not attached to Neptune is because they are entrained in the peculiar anti-aligned secular dynamic with Planet Nine.

Finally, there is the puzzle of the highly inclined orbits. Whenever one sees cycling of orbital inclination and eccentricity, there is a temptation to invoke the Kozai-Lidov mechanism as the answer. In the case of Planet Nine, however, the high-inclination dynamics are keenly distinct from those facilitated by the Kozai-Lidov effect. Perhaps the most obvious reason why the dynamics we observe in numerical simulations is not the Kozai-Lidov effect is that in our calculations, highly inclined KBOs develop the highest eccentricities when their orbits are perpendicular to the plane of Planet Nine’s orbit, in direct contrast with perpendicular and circular orbits entailed by the Kozai-Lidov effect.

So if it’s not the Kozai-Lidov resonance, then what is it? As it turns out, the high-inclination dynamics induced by Planet Nine is characterized by the bounded oscillation of a octupole-order secular angle which is equal to the difference between the longitude of perihelion of the KBO relative to that of Planet Nine and twice the KBO argument of perihelion. How could we have ever thought it was anything else?… The plot below shows the high-inclination secular resonant trajectories executed by test-particles in our simulation plotted in canonical action-angle coordinates, with the observed objects shown in orange. Examining this plot closely, one detail that I’m reminded of is the fact that the few high-inclination large semi-major axis centaurs that we know of are actually the “freaks” of the overall population, since they all have perihelia on the order of ~10AU. Certainly, detecting a sample of these objects with perihelia well beyond 30AU would be immensely useful to further constraining the parameters of the model.

With the above rambling in mind, I will admit that all I’ve mentioned here is an introductory account of the paper. As such, it represents a considerable simplification of our actual calculations, so if you want to better understand the full picture, I can only urge you to read the paper itself. Importantly, however, the work presented in this manuscript not only provides us with a better understanding of how the observed census of distant KBOs has been sculpted by Planet Nine, it finally places the P9 hypothesis within the framework of Popper’s demand for falsifiability, and sincerely allows for the confrontation of the Planet Nine theory with the observational search. The final step is now to find it.

I'm curious; do you know what the albedo of Planet 9 must be in order for you to detect it? If for some reason it is very dark and reflects almost nothing, would it be possible to detect it by checking if any stars are eclipsed?

ReplyDeleteThis comment has been removed by the author.

DeleteWe typically assume that it has a similar albedo to that of Neptune. There is reason to believe that it is in fact much *more* reflective (as pointed out in this paper: https://arxiv.org/pdf/1604.07424.pdf) If it is really dark, we would have to go out to higher magnitude. Occultations of stars, however is unlikely to be a viable avenue for discovering P9.

DeleteThank you for keeping us up to date! It's amazing to follow your search. I know it's a bit early to know, but how thin or thick do you think the atmosphere will be? Given that it's supposed to be less massive than Uranus and Neptune, how different could it be?

ReplyDeleteThere is really no way to know for sure. But my bet would be on a ~10% H/He envelope fraction - consistent with typical extrasolar super-Earths.

DeleteDo we know much about atmospheres beyond the solar system? I was under the impression that that's the sort of thing we'll need James Webb for.

DeleteCould a secular or other interaction between P9 and the 30 Earth masses of Neptune-scattered KBOs be responsible for raising P9's perihelion?

ReplyDeleteIn principle, absolutely. This is not something we've looked into in detail yet.

DeleteHi Mike and Konstantin. I think,...I also calculated,..found from simulations and other sources 15 years ago that there should be Planet,...on orbit with period cca 20000 years. You proposed, that that one planet should be cca 10Earth masses. What is but exact limit for that mass? Is it 8-20,..or 5-100 Earths masses,...? Did you invoved into sim. also influence of Alpha Centauri, for to find mass of P9,10....?

ReplyDeleteNo - the influence of Alpha Centauri is not in our simulations...

DeleteIf we consider that gravitational influence of Alpha Centauri stars on our planets is cca like 1/2Earth mass in distance of supposed P9,10,..(300AU),..it is not too much

Deletelittle more precise with calculator,...gravit. influence of all 3 Alpha Centauri in our Solar system is equivalent to 1 Earth mass body placed to distance 346AU-2Ldays-cca distance of P9,10,... Pavel Smutny

DeleteI dont know if you think about motions of barycenter for our Sun...there should be amendment of each planet proportional to its mass and to distance fron Sun.....

DeleteUnknown: I think you've calculated the gravitational acceleration caused by Alpha Centauri (which is proportional to the inverse square of distance). But this effects all Solar System objects nearly equally. If you want to know how the Solar System orbits are influenced, you'd need the tidal acceleration (proportional to the inverse cube of distance).

DeleteLittle more about motions of Sun,..barycenter,..Maximal distance of Sun from barycenter is 4,4 radiuses of Sun. If only Jupiter is there (in our solar system), so 1 radius of Sun should be there,,,Amendment of Saturn could be till 2/3 of 1 radius,...Uranus, Neptune,..could give cca 3/5 of 1 radius,... together cca 3 radiuses,..but where is the rest,.._? Amendment, shift of barycenter due to P9,10 could be visible ,..especially after longer time,...Pavel Smutny

DeleteIn your simulations how much does the inclination of the orbit of the perturb in P9 planet change as a result of the interactions? In the course of altering the inclination (Momentum) of the solar system by about 6 degrees, P9's orbit must also be altered (?) What would the change look like over time and might you be over-estimating the present inclination because of the its history of a higher inclination? Would the scattering reflect more of the historical average inclination than on the contemporary one?

ReplyDelete-jdk

Thanks for the ongoing communications.

You're absolutely right, the inclination of P9 (with respect to the ecliptic) changes, too (because the ecliptic changes). We fully track all of that for our current inclination predictions. Sadly, our current inclinations aren't as precise as I would like, so in the end it doesn't make all that much difference.

ReplyDeletectrl + is my friend.

ReplyDeleteDo the objects move clockwise around the trajectory in Fig. 4?

ReplyDeleteThey sure do.

DeleteI would like to see a more practical visualization of how the orbits of the high inclination objects evolves.

ReplyDeletePerhaps a plot of the location of the perihelion of one of the high inclination objects plotted in ecliptic longitude and latitude, with arrows showing the direction it travels in its orbit to indicate the orbit's orientation, possibly scaled by eccentricity would work.

This could also provide predictions of where they should be observed, in the interest of falsification.

Better yet, check out the orbits in physical space on the second figure of the very first post: http://www.findplanetnine.com/2016/01/premonition.html

DeleteAfter reading in the paper that the longitude of perihelion is a dog-leg angle I recalled that in the first paper latitude and longitude was used in one of the figures instead. That and Fig. 10 left me wondering what sort of path the perihelion (and the orbital pole) follows. Would they follow circles if plotted on a globe or a more complex curve?

DeleteIn the initial 2016 papers a range of a9=200-2000AU, e9=0.1-0.9, m9=0.1-30Me was explored with w=0,180. When looking at the most up to date trans-Neptunian orbits (those highly aligned for a>250AU), there seems to be a small void (little to no orbital overlap) for small a9 near w=0. More specifically, it is quite apparent for all angles within 0 < w < 90 and 270 < w < 360, only a small angular window near w=0 is cleared out along the ecliptic. Your simulations included w=0, a9=200AU, e9=0.9, m9=10Me, which would roughly appear to fit within this region. What are the secular versus numerical results of these calculations? In particular, what is revealed for plots of: 1) e vs. delta(w) for different values of a, and 2) e vs. delta(w) for different MMR?

ReplyDeleteIntriguingly, the secular dynamics inside MMRs mirrors the purely secular dynamics (where the orbital motion of P9 and the KBO is assumed to be uncorrelated) at the corresponding semi-major axes rather well. I note that formally, this purely secular dynamics is inapplicable when orbits cross because of emergent singularities in the gravitational potential. Remarkably, however, these singularities are integrable allowing one to essentially get the right answer while being formally out of bounds of where the theory is supposed to work.

ReplyDeleteIs the integral in equation one of the paper determined analytically or numerically?

DeleteAs ever, thanks for the time and effort you and Mike expend on explaining what's going on to we blog-followers, Konstantin.

ReplyDeleteYou wrote:

"So if it’s not the Kozai-Lidov resonance, then what is it? As it turns out, the high-inclination dynamics induced by Planet Nine is characterized by the bounded oscillation of a octupole-order secular angle which is equal to the difference between the longitude of perihelion of the KBO relative to that of Planet Nine and twice the KBO argument of perihelion. How could we have ever thought it was anything else?…" ðŸ˜‚

You're in good company ;-)

"My reflection, when I first made myself master of the central idea of the ‘Origin’ was, ‘How extremely stupid not to have thought of that!’"

(Huxley, Thomas Henry. 1887. On the Reception of the ‘Origin of Species,’ in Darwin, Francis (ed.). 1887. The Life and Letters of Charles Darwin, Including an Autobiographical Chapter (London: John Murray), volume 2, pp. 179-204. Quote is on v2:p197. View original page here:

http://www.ucl.ac.uk/sts/staff/cain/projects/huxley/how_extremely_stupid)

!!!! Whoa...

DeleteIs data analyses from your last Hawaii trip over or there is still possibility that it is already found? :)

ReplyDeleteBarely started. I give it 15-20% chance that we found it in this run.

DeleteIs the production of the high inclination objects connected to Planet Nine's inclination relative to the Solar System, its eccentricity, or a combination of the two?

ReplyDeleteOh, wait. The secular angle wouldn't include the difference between the longitude of perihelion if Planet Nine didn't need to be on an eccentric orbit to produce perpendicular orbits.

DeleteIt's a combination of both. You are correct that this effect simply does not exist for circular angles, since P9's longitude of perihelion is ill-defined.

DeleteDoes the number of perpendicular objects vary with Planet Nine's inclination?

Deleteyes it does!

DeleteEnough that it can be used to constrain Planet Nine's orbit?

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DeleteWe know lot of planetoids on prolongated eliptic orbits.There is group like on:https://upload.wikimedia.org/wikipedia/commons/2/2a/Planet_nine-150au-etnos_now-close_Jan-2017.png which have main axis cca 150AU, what is also in good coincidence with Titus-Bode Law. Closer P9 should have some such orbit, orientation. P10-could be on orbit like Brown with Batygyn foretold too. The second, ore distant group like on:http://www.nature.com/news/planets-graphic-jpg-7.33255?article=1.19182 could be in other resonance ratio but with P10,... or are those orbits such only due to projection of relative motions of planetoids towards Sun,..known planets, due to much more massive P10,P9,... than oficially foretold,... Pavel Smutny

ReplyDeleteIn the paper its stated that 38% of all stable objects experience at least one high inclination excursion.

ReplyDeleteLooking at Fig. 10 it appears that the excursions last 1.5 - 2 Gyr. Would roughly 15% undergoing one of these at a particular time be accurate?

That's correct - excellent insight!

DeleteAfter I posted that I thought it might be higher, for example if a high inclination object was more like to be lost when its perihelion was less than 30 AU.

DeleteI am a high school student with a recently developed intense interest in astronomy, can you possibly answer these questions of mine please? If Planet Nine exists, but proves too difficult to detect with Earth-based telescopes, would it be possible for NASA to divert the New Horizons probe to the likely location of Planet Nine? Would that consume too much fuel? Would it be impossible due to no remaining large gravitational wells for it to slingshot around? Or would it be more likely that due to the core radiation likely emanating from said hypothetical planet, the infrared telescope of James Webb could detect the planet right off the bat? If you read this or answer this, Mr. Batygin, thank you.

ReplyDeleteNew Horizons is traveling very quickly into the opposite direction, so no chance. James Webb would be immensely useful for *characterization* but not detection. It's not a good tool to "look" for stuff.

DeleteThis comment has been removed by the author.

DeleteI'm not sure if I understand your definition of "characterization."

DeleteDetection of its atmosphere, heat radiated, possible moons would be my guess.

DeleteIf the James Webb infrared telescope could hypothetically detect faint brown dwarves dozens of light years away what would prevent it from detecting a planet only 500 AU out or less? One light year is over a thousand times that. Surely if the planet were there a modern infrared telescope without any atmospheric interference could, in your words, characterize it?

ReplyDeleteSurely its revolution speed even near aphelion would be significantly apparently faster than the background stars through parallax alone?

The James Webb Telescope's field of view is too small, a few arc minutes across, to be used for searches.

DeleteNow this looks interesting:

ReplyDeleteCircularizing Planet Nine through dynamical friction with an extended, cold planetesimal belt

https://arxiv.org/abs/1710.08295

They find better odds of reproducing the proposed orbit (20-30%) than some others I've seen, though I wonder about their starting point with P9 perihelion at 30 AU.

Yes - thanks for pointing it out. It's a really cool paper, and may point towards the potential existence of a cryobelt in the primordial solar system. Very cool set of calculations.

DeleteRegarding P9, is it possible that there could be P10, P11, P12.. ?

ReplyDeleteHow about taking this even further: could there be tens, or even hundreds of planets, orbiting at distant 100+ AU orbits, or even 1000+ AU orbits? If they existed, we'd have no way of detecting them now of course..

I never understood why that space should be empty of planets. Is there something in the theory of solar system formation that precludes planets at such large orbits? Note that the distance to Alpha Centauri is 276,173 AU. So even, say, a 2,000 AU orbit would be still be very close to our Sun -- only 1% distance to Alpha Centauri.

And.. how about taking it even further.. is it possible that there are planets that were ejected from either our system or that of Alpha Centauri, and that orbit around both Sun and Alpha Centauri, in a 3-body system, at some massive distances out there in interstellar space? Is there anything in known physics or celestial mechanics that provably precludes this?

First of all, whether there will be more planets depends on the planet definition. If you go by the IAU definition, then Planet 9 will just qualify to be a planet, but for example "Planet 10" proposed by Malhotra and Volk (on an orbit between Neptune and P9) would not - even though they think it has about the size of Mars, it would not be massive enough to clear its orbit, it just leaves an imprint on the inclination of the orbits in its vicinity.

DeleteIf P10 exists, there could be more objects like it, but objects fulfilling the IAU definition are much more unlikely - the further out they are, the more massive they need to be to clear their orbits, but the more massive they are, the less likely it is they were ejected by Jupiter and Saturn in the first place.

Onwards to your last question: planets that are ejected are pretty much gone for good. They either orbit the Sun, or they don't. The next thing they could orbit is the Galactic Center. Once a planet gets about 4 light years away from the Sun, then the tidal forces from the Galactic Center would pull it even further away, and from then on Sun and planet will be on different orbits around the Galactic Center, and they will never meet again.

Alpha Centauri is only near the Sun right now - 100000 years in the past or future it would be light years further away (but still visible), and millions of years in the past or future it would even disappear from view.

(Alpha Centauri is interesting for another reason - as a type of system. If a planet was ejected from e.g. the orbit of Alpha Centauri A, then there would be the possibility that it orbits in the Alpha Centauri system as the collective gravity of A and B together could be strong enough to keep a planet that escaped from A)

I'm trying to visualize how and why the orbits of anti-aligned objects evolve the way they do:

ReplyDeleteWithout Planet Nine the large semi-major axis would precess round and round without any variations in their eccentricity.

But when P9 is added an object that starts anti-aligned will see P9 more and more often ahead of it than behind as delta longitude of perihelion increases as it precesses. This causes its angular momentum to increase which decreases its eccentricity.

As its eccentricity increases its precession rate slows. The orbit of P9 is also precessing and it starts to catch up that of the smaller object decreasing delta longitude of perihelion.

When delta longitude of perihelion drops below 180 degrees the object sees Planet Nine behind it more often than ahead of it. It begins to lose angular momentum and its eccentricity decreases reversing the process.

When its eccentricity is again large its precesion rate increase as does delta longitude of perihelion and its orbit returns to the staring point.

Is this about right?

Oops, this explanation: 'will see P9 more and more often ahead of it than behind' doesn't work with the aligned objects. Guess I oversimplified that.

DeleteHi,..maybe ancient Nordic cosmology myth concerned with Yggdrasil=World Tree can help. There are snow giants on the upper part of this World Tree= Our Solar System,...look on my web: http://www.hamops1.szm.com/ Pavel Smutny

ReplyDeleteCosmic rays technology successfull at discovery of new two chambers in The Great Pyramid in Gisa.

ReplyDeletehttps://www.youtube.com/watch?time_continue=445&v=ZB-MOGw0RMo

My discovery, findings some years ago show that pyramid complexes in Gisa were used for receiving, transmitting cosmic rays,..for to detect,....P9,P10,..http://senmut.beep.com/narmeratlanteantech.htm Pavel Smutny

Kind of like in evolution of life, to perform Natural Selection, first, you need to generate, by some self-organization, a variety of lifefors. To perform falsifiability, one first needs to have a theory.

ReplyDeleteI personally like Jacob Bronowski's "Origins of Knowledge and Imagination." I"ve made some generalizations of his ideas in the mentioned book. I'll just copy and paste one of my last write ups about it,

"People like to think that what they see in terms of visual senses is what the world is really like. James Maxwell's elucidation of the electromagnetic nature of light is enough to dispel that notion. Maxwell's discovery showed that light comes in many other wavelength's as well - infrared, ultraviolet, x-rays, and gamma rays. Maxwell's discovery was the greatest physics discovery since Isaac Newton; but, getting back to the point. We cannot get at nature directly. We have to infer nature indirectly.

Our eyes see with discreet rods and cones that can only see some wavelengths of light. We've had to use facts and logic to straighten out our conditioned perspective. We have to establish boundaries, or error rates for which things apply or not. Scientists use terms like accuracy and precision of measurements. Accuracy refers to how accurate your measurement is to a known or theoretically calculated quantity. Precision is how close each measurement is to one another. Mathematicians have a reversible definitions idea for mathematical definitions. If one can reverse the definition, and mean the same thing, then the concept is well defined. Jacob Bronowski argues, and I agree with him, in his, "Origins of Knowledge and Imagination", that the universe is this infinitely detailed whole. Any cut in it can only be an artificiality. But, clearly, mathematicians have found that one can restrict the field of applications of a concept to a certain time and place. Of course, in time, as structures, or concepts, influence one another in this energetic universe, things influence one another, and one has to revise, or generalize concepts. Getting back to restricting ideas in a certain time and place . . .

Jacob, once again in his 'Origins' book mentioned above, suggests that when one makes a cut, establishes a boundary, for however fleeting a moment of applicability, certain undefined terms fall out. He considers concepts like inertia, electron, atom, the sun centered solar system really, are of this nature. We didn't first build a spaceship, launch it to some polar position, look down, and say, "look, the sun is the center of the universe." As described above, we decoded it out of noticing that the Sun seemed abnormally large compared to the other 'planets'. The phase of the moon seemed to point to the Sun. And, the planets seem to precess; they stop their forward movement, loop backward, and then move back forward again. What one does is dam up nature - like a hydro-electric dam. And in so doing, one generates electricity. I've noticed an analogy between Jacob Bronowski's ideas in his 'Origins' book and some connections ideas in James Burke's 'Connections' book and video series.

"In James Burke's Connections(and this is an idea in archaeology and anthropology long before), agriculture leads to technologies. Clearing the fields(I mean like ‘slash and burn’ to remove everything except the plants desired to grow) and having to farm introduces one to problems of irrigation, pest control, and fertilization. I'm equating these to the new undefined terms that Jacob Bronowski is pointing out above. Also, agriculture, once it gets established frees up the populace to do other jobs, or ideas. Jobs/Ideas . . . like a military, a scribe to keep track of taxes, business people, priests, rulers, and mathematicians. The farming is predicated on the seasonal cycles. So, one needs a calendar, and an astronomer and mathematicians.

DeleteJames Burke notes some more things in episode/chapter two and three. In episode 2, Guericke is found producing the first vacuum, and doing all sorts of amazing things with it. He keeps two hemisphere's of a ball together by a vacuum. A vacuum so strong that two horses can't pull it apart. He then notes that Guericke presents his vacuum to a German prince. And he shows that mice suffocate in it/bell's don't make a sound when you try to ring it/fire goes out. In episode 3, he shows how the horse takes center stage for all kinds of technologies and is the basis for the second half medieval Europe economy. There's the horseshoe, the neck brace for the Horse, and the stirrup and saddle for the Horse rider. Then, there's the Knights armour. But, these are all very concrete technological ideas. What's the connection between the concrete technological world and abstract mathematical ideas such as number, one-dimensional lines that don't really exist beyond our heads?

I'd suggest as Susanne K. Langer, in her "Introduction to Symbolic Logic" does, the difference between constituent relations and logical relations. Constituent relations are like the verbs in a sentence. Logical relations are like conjunction/disjunction/inclusion of deductive reasoning. These are similar relations to set theory relations of union/intersection and equality of sets. And, as mathematicians noted in the eighteen hundreds, numbers are like equivalent sets.

Numbers are the abstract identity of equivalent, or analogous sets - apart from there concrete manifestations. Like there's a couple of apples, and a couple of oranges. The apples and oranges are concrete manifestations of the abstract concept of number two. Everything has a structure. a Relation with parts that makes sense with respect to that relation. There's binary relations, and relations that can take on three parts, or terms. For instance, between is a triple relation because something is between two other things. You would never say ‘love between john’. But, within a given relation, say Jane loves John, there's a whole set of terms, or parts, that makes sense with the love relation. You could even say Ferrari loves racing. But, taking the relation, the binary form from all it's concrete manifestations is what Symbolic Logicians and Mathematicians call an abstraction.

When we present a specific instance of a concept, we're thinking of it in a very specific context. If I just mention some verb like running, you think of either Paul is running away from something. Or Paul is running the show. Something very time specific. Only by noting the general concept, does the concept go beyond the very time specificity of this or that structure. When humanity first started noting ideas, the ideas were very provincial. The number two was associated with male and female properties. As ideas are generalized, they strip themselves from imprints of their origins."

Can we expect any shiny new frozen planets under our tree for this holiday? Or maybe some witty new ideas and math to bring good cheer?

ReplyDelete