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Posts tagged with posets

‘The Descent of the Modernists’, cartoon by E. J. Pace, first published in Seven Questions in Dispute by William Jennings Bryan (1924).
with some nice autobiographical details, via cmnotes




A sensible partial order ranking countries in the 2008 Olympics
—made by the guy who wrote PuTTY
I love posets because of situations like this. Not everything in life is completely one-dimensional, but that’s not to say it’s 100% disorderly either! Mathematics has some pretty good ideas for how to rationally label the world, sometimes.
  
Tatham’s description of the dilemma of ranking countries absolutely on 3 criteria reminds me of
multicriteria modelling
Arrow’s impossibility theorem
various multicriteria bargains or decisions to be made in regular life, like
which house to buy
which job to take
which city to move to
which président to elect
which school to attend
which product to recommend to a customer
all of which, if they’re quantifiable at all, usually consist of multiple independent, uncombinable factors (Would you like your président to be humane, or good with the economy? Well, I’d ideally like both… Would you like your employer to not beat you, or to pay you a living wage? Um, again, ideally both…). In the most naïve approach are combined with a linear weighting. (A concave weighting with positive cross-partials should be more sensible.)
As Tatham notes, using the magic of irrational numbers it’s possible to guarantee uniqueness of a ranking—but would the ranking be any good?
Well given that we know K Arrow’s Impossibility result, maybe we should just reduce our expectations. Instead of trying to squeeze a 2-ton elephant into a miniskirt made for Kate Moss, maybe we should relax the requirements and just hope to get a poset. That can be more doable.
It’s an application of Vilfredo Pareto’s genius idea.

A sensible partial order ranking countries in the 2008 Olympics

—made by the guy who wrote PuTTY

I love posets because of situations like this. Not everything in life is completely one-dimensional, but that’s not to say it’s 100% disorderly either! Mathematics has some pretty good ideas for how to rationally label the world, sometimes.

  

Tatham’s description of the dilemma of ranking countries absolutely on 3 criteria reminds me of

  • multicriteria modelling
  • Arrow’s impossibility theorem
  • various multicriteria bargains or decisions to be made in regular life, like
  • which house to buy
  • which job to take
  • which city to move to
  • which président to elect
  • which school to attend
  • which product to recommend to a customer

all of which, if they’re quantifiable at all, usually consist of multiple independent, uncombinable factors (Would you like your président to be humane, or good with the economy? Well, I’d ideally like both… Would you like your employer to not beat you, or to pay you a living wage? Um, again, ideally both…). In the most naïve approach are combined with a linear weighting. (A concave weighting with positive cross-partials should be more sensible.)

As Tatham notes, using the magic of irrational numbers it’s possible to guarantee uniqueness of a ranking—but would the ranking be any good?

Well given that we know K Arrow’s Impossibility result, maybe we should just reduce our expectations. Instead of trying to squeeze a 2-ton elephant into a miniskirt made for Kate Moss, maybe we should relax the requirements and just hope to get a poset. That can be more doable.

It’s an application of Vilfredo Pareto’s genius idea.

(Source: godplaysdice.blogspot.com)


hi-res




It’s wrong to say that faith and science are opposites,

  • not only because that’s playing into the presentist viewpoint of American fundamentalists fighting to teach creationism in science class versus /r/atheism, but because
  • scientists don’t choose their research programmes at random. They “have a hunch” — or an aesthetic sense impels them. But staking your career on  the belief that a particular line of investigation will be fruitful, both in a scientific sense and in a value-to-humanity sense, requires stronger language than merely “I think so” or “I have a hunch”. I think it’s fair to say that scientists have faith in their research programmes.

I’ll give an example of a research programme that I have faith in. Mostly unjustified faith, but I believe it nonetheless. (I could be wrong, of course — but still I can’t approach the world with no beliefs whatever — although some views of rationality would suggest that this unlivable mental life would be the most honourable way to live.)

  1. I believe there’s something wrong with economic theory. Call it a dark age on the way to enlightenment, call it an obsession with equilibrium-and-optimisation, call it the undue influence of Milton Friedman essays on the deeper, unspoken beliefs of economists vis-à-vis effect of careful studies or creative mathematics. ∃ many ways to describe the malaise | muddle | distraction | not even really sure what to call it.

    This is not based on "Economists didn’t foresee the financial crisis!" or a critique of the Washington Consensus. It’s not about Objectivists or people who don’t understand what a model is, but rather at real, non-crazy economists. It’s more based on statements like "Economics is in a terrible state"—Ariel Rubinstein. Or questions like: since information and search costs and other such things dominate the f**k out of the normal incentives-based thinking we use to armchair-speculate—then what is even the use of the partial-equilibrium intuitions or DSGE or anything like that?

    I also don’t think this idea would necessarily change the focus to more sociological or historical or cultural issues (like economists ignoring how utility functions come to be, or larger questions about history and culture and family norms … I actually think a lot of economists are already prepared to focus on those issues, they just need to make them mathematically tractable). Rather my gut instinct tells me that this research programme is “far upstream”—redirecting the river by diverting the water long before it becomes a rushing channel (sometimes called the MSNBC channel) that’s too powerful to redirect.
    nerve of cover
  2. I don’t know enough about sheaf theory or cohomology to say for certain whether they can be used for this or that. It’s just my spider sense tingling when I look at the ideas there. Most of the applications I’ve read about are to either physics problems or logic, or to higher mathematics itself (algebraic topology, algebraic geometry, topological analysis, … stuff that’s named as (adjective = way of thinking + noun = subject matter)).
    presheaf axioms
    That said I think there’s something to be found here in terms of new viewpoints on economic questions.
  3. Consider the Leontief input-output matrix (Cosma Shalizi recently wrote a lot about it in his book review of Red Plenty on Crooked Timber blog).
    hypothetical transactions from a leontief input/output matrix
    Mathematically savvy people know that every graph can be encoded as a matrix, and furthermore with the right base corpus and some knowledge of “characters” we can do one-directional graphs.
    social graph
  4. What’s the [putative] application to economics? Well instead of thinking about all this stuff we can’t observe or interpret yet—utility curves, willingness-to-pay outside the lab, valuations, etc.
    indifference curves
    (we don’t even know experimentally if there is such a thing as a valuation—and it’s kind of dubious—yet we go on as if these things exist because they’re axiomatic keystones of the only tractable theory around). Instead of continuing to rely on the theoretical stuff handed down from Bentham, let’s think about all the things we can measure—like transactions—and ask how we can use mathematics to make theories about those things and possibly infer back to the stuff we really want to know, like is capitalism making the world a better place.
  5. Transactions are one place to start. Prices (like the billion price project) are another. And the web now generates huge amounts of text—maybe we can do something with that. But let’s start by going back to the Leontief matrix.
    input output matrix
  6. In the formulation I learned in school, there’s a fixed time unit—like a year—and each dimension corresponds to an exactly comparable item class—so like a three button shirt and a four button shirt would be separate dimensions, but once we finally get down to a dimension, everything along that dimension is equivalence-classed.
  7. I can see three things missing from that picture.

    First of all, I want to be able to “zoom in” to different timescales and have my matrix change in the sensible way. In other words I want a mathematical object that operates on multiple timescales at once, with a coherent, consistent translation between the Leontief matrix of October 17th between 19:29 and 21:13 GMT, and the Leontief matrix of 1877 A.D. I believe things floating around sheaf theory are the place to look for that.
  8. Second of all, I want neighbourhood relationships (and even distances) between the items—so that a three-button blue blouse is “closer to” a four-button blue blouse than it is to a ferret named Bosco the Great sold at the Petco in Moravia, Illinois. So something from algebraic topology is necessary here.
    algebraic topology
    Maybe a tie-in to “lumpy human capital”—the most important kind of good because it’s what humans use to sustain themselves and help others. It’s acknowledge to be “lumpy” in that ten years of studying economic theory doesn’t prepare you to be a laundress or even necessarily to trade OTC derivatives. But we also know that in terms of neighbourhood relationships, economic theory studying is “closer to” finance than to farming. (Although most economists are not as close to finance as seems to be generally thought.)
  9. Both of those two points are more just æsthetic problems or issues with foundations. Like philosophical gripes could be solved, in the same way that a transition from cardinal utility to ordinal utility, even though I don’t think the outcomes of the ordinal utility theory were very different.
  10. Third, I want my matrix to be time-varying or dynamical. New trade partners come into existence, some businesses shutter their doors and file their dissolution papers, others are broken up and sold in parts, and even with an existing vendor I am not going to do the same business each year. Some of these numbers are available in XBRL format because public companies sometimes do business with each other.
  11. Fourth, and here is where I think it would be possible to get new ideas of things to measure. If I have some kind of dynamical, multi-level, “coloured” graph of all the trades in all the currencies and all the goods types in the world over the right number corpus, then I have a different mathematical conception of the world economy.
    a drawing of some foreign exchange flows I did whilst reading about cohomology ... this is the beginning of some ideas which are both graphical and hard to draw (because they're so convoluted).).
    I can draw boundaries like you would see in a cell complex and denote “a community” or “a municipality” or “a neighbourhood” or “a province” and when I perturb those boundaries some rationality conditions need to hold.
    cells with directed boundary
    Taking this viewpoint and applying only the maths that’s already been invented, people have already found a lot of invariants on graphs—cohomology invariants, generalisations of Gauss’ divergence theorem, different calcula on the interesting objects (like fox calculus)—and applying those theories to the conception of the super-duper Leontief matrix, we might find new things to measure, or new ways to make different sense of some measurements we already have.
    fox calculus
    If you remember this Perelman quote about calculating how fast Christ would have to run on the water to not sink in, or various nifty cancellations in the vacuum states of a gnarly physics theory — that is the kind of thing I’m thinking could be useful in theorising new invariants to measure from an überdy-googly Leontief trade matrix.
    string diagrams for the Frobenius algebra axioms

    Or from www.math.upenn.edu/~ghrist/preprints/ATSN.pdf we learn "The Euler characteristic χ of any compact triangulable space is independent of the particular [finite] simplicial structure imposed, as well as independent of the topological type.”
    ok, not a triangulated space. but this gives you the idea.
    Yum. Tell me more.
    another drawing from the first time I was reading about cohomology and thinking about its possible applications to a dynamical graph of actual transactions. Looking back on it this isn't super clear either. Maybe you get what I was "suggesting" though.
    For example we know some Gaussian-divergence ∑ relations that happen within the grey box of a firm—all the internal transactions have to add up to what’s written on the accounting statements. But what about applying this logic to a group of three firms that circularly trade with each other and also each has a composite edge (with different weights) adding up all of their trades to “outside the cycle”?

    Seems like some funky abstract nonsense could simplify problems like that and, crucially, tell us invariants that give us new ideas of what to measure.
  12. Fifth, this is not really related. I think the concept of symplecticity from physics nicely captures the essence of what tradeoffs are about.
    a symplectic manifold. drawn by r ghrist in the /preprints/ATSN.pdf
    But I’m still looking into this—I won’t definitely say that, it just seems like another fruit-lined avenue. 
  13. There are tie-ins to categories, causal diagrams, and other stuff wherein I may be just lumping together a lot of seemingly-related ideas.
    A causal diagram. (not to be confused with an acausal diagram!)
    So I’m not sure if looking at a super-duper Leontief matrix like described above would have nice tie-ins to causal graphs / structural equations à la Judea Pearl, but hey it might. At least one tie-in I can already think of is that all the goods actually transacted doesn’t tell you enough because there are threats and possible counterfactuals and CV’s that are sent in but get ignored or rejected, or smiles and pats on the back which are a kind of transaction that influences the economic outcomes without being tied directly to money or a goods transaction.
  14. Why go for even more abstraction, even “more” maths, when so many of the critiques of economics say it’s become too mathematical? Simple answer. More abstract mathematics requires fewer assumptions. So conclusions drawn using those tools are more likely to actually hold true in the real world. For example, is it more plausible that someone’s utility increases linearly, or monotonically with good X? Monotonically of course is much more realistic, although we could infer much more if linear were the case. But what’s the point of making easier inferences if they’re wrong because the assumptions don’t hold? Hence the interest in more general, more abstract mathematics.

Now, realistically? The scale of investigating this “hunch” in terms of concrete steps that lead A → B → C → D are way beyond what I will probably accomplish. Even if I dropped all side interests and all work, it would take at least a couple years to get publishable material out of these hunches.

But that’s exactly my point about science. I was told by a Zaazen practitioner that this is kind of a Zen-like paradox.a reflexive node (CYCLIC graph, not a dag) In order to investigate the premise that there are useful applications of sheaves & cohomology to economic theory, I first have to accept the premise that there are probably useful applications of sheaves & cohomology to economic theory.


cylical graph / circular logic

Glancing at the text above you can probably tell that my thoughts on this issue are formless, probably mischaracterising the mathematics I’ve only heard about but don’t yet understand. My mental conception of these things, if it could be understood via a perfect future theory of mental representation and fMRI snapshots of my mind thinking about this stuff, would be some mixture of formless and inaccurate.

So the important decisions (decisions of major direction, not adjustments or effort) are made amongst the formless, but can only be harvested as a form. Like the beginner’s mind, with its vagueness and formlessness, giving way to the expert’s mind, its definition, choateness, and exactitude. (Form and formlessness being complementary in the QM | vNA sense.) I think that’s Zen as well.




The word < is normally defined to mean less than in some quantifiable sense. For example, considering the set {3,6,1441}, one could say that 3<6<1441.

But in the abstract language of partially ordered sets, < is reinterpreted many ways — to mean proper subset of ⊂  (contained by), divides, “is hotter than" or … any transitive relation — even begat.

Consider the set

  • {Cain, Enoch, Irad, Mehujael, Methusael, Lamech₁, Jabal, Jubal, Tubal-cain, Naamah} ∪ {Adam, Abel, Seth} ∪ {Seth, Enosh, Kenan, Mahalalel, Jared, Enoch₂, Methusaleh, Lamech₂, Noah} ∪ {Noah, Shem, Ham, Japheth}.

Transitivity means that it’s impossible for Adam < … < Adam < … (where Adam refers to the same man, not to another person also named “Adam”. We can call him Adam₀ if it’s a problem).

Then the fourth and fifth chapters of Genesis yield the following relations among the members of that set.

  • Cain < Enoch₁ < Irad < Mehujael < Methusael < Lamech₁ < Jubal
  • Lamech₁ < Jabal
  • Lamech₁ < Tubal-cain
  • Lamech₁ < Naamah
  • Adam < Cain
  • Adam < Abel
  • Adam < Seth
  • Adam < Seth < Enosh < Kenan < Mahalalel < Jared < Enoch₂ < Methuselah < Lamech₂ < Noah
  • Noah < Shem
  • Noah < Ham
  • Noah < Japheth

It’s not like we reduce Enoch₂ to “the thing between Jared and Methuselah” — there is other information attached to Enoch₂ such as that he walked with God and was no more (whereas all the others were noted to have died). Likewise to say that 18 is the integer between 17 and 19 isn’t to ignore the fact that 6 divides 18 or that it represents a legal bright-line in some countries.

Nor do we assume that every pair (a,b) from that set should be comparable. (In a totally ordered set either a<b or b<a, ∀a,b.) But in case of the begats:

  • Mehujael Noah
  • Shem Tubal-cain
  • Jared Jabal




It’s not always possible to say A ≻ B or A ≺ B. Sometimes

  • neither A nor B is smaller.        A≹B
  • neither A nor B is more successful.   A≹B
  • neither A nor B is prettier.         A≹B
  • neither A nor B is smarter.        A≹B
  • you don’t love A any more or any less than you love B.   ℒ(A)≹ℒ(B)
  • neither A nor B is tastier.           A≹B
  • neither A nor B is closer.           |A−x| ≹ |B−x|
  • neither A nor B is more fair.        A≹B
  • neither A nor B is better.             A≹B

I’ve argued this before using posets. And I intend to argue it further later, when I claim that the concept of Pareto superiority was a major step forward in ethics.

*[The concept of Pareto dominance allows you to make, at least in theory, a valid, fully general comparison between two states of the world. A≻B in full generality iff   a ≻ b   a ∈ A and ∀ b ∈ B, by the individual standards of ∀ .]

 

For now, though, I’ll draw some examples of functionals that don’t beat one another. That is, ƒ≹g nor g≹ƒ. (You might assume  has to be 2-symmetric but I’m just stating it for clarity.)

In this drawing, green wins sometimes and purple wins other times. Is it more important to win the “righthand” cases or the “lefthand” cases? How much better for each scenario? (see integrating kernel) Is it better for the L₂ norm to be higher? Or just for the mass to be greater?

In this drawing, orange wins sometimes and blue wins other times. Is it more important to win the “interior” cases or the “extremal” cases? How much better for each scenario? (see kernel of integration)

 

How about a function that measures the desirability of a particular boyfriend / girlfriend in various scenarios. How about the function g measures boyfriend B in the various scenarios (domain) and the function ƒ measures boyfriend A in the various scenarios. By measures, I mean the function’s codomain is some kind of totally ordered set where it does make sense to talk about better ≻ and worse ≺.

  • ƒ(at dinner) ≻ g(at dinner)
  • ƒ(career) ≺ g(career)
  • ƒ(in bed) ≫ g(in bed)
  • ƒ(with your family) ≺ g(with your family)
  • ƒ(at the beach) ≺ g(at the beach)
  • …and so on…

So how do you decide whether A≺B or B≺A? Perhaps you have your own priorities sorted so well that you can apply a kernel. Or perhaps AB in the final analysis.

I could make a comparable list for

  • comparing two houses or apartments (well, this one’s closer to the park, but that one has that cozy breakfast nook),
  • comparing two societies (one where the top marginal tax rate is 41% and one where the top marginal tax rate is 40%),
  • and on and on.

Sometimes it’s hard to compare. Sometimes — like which of your kids do you love the best — it’s impossible to compare.




Paul Finsler believed that sets could be viewed as generalised numbers. Generalised numbers, like numbers, have finitely many predecessors. Numbers having the same predecessors are identical.

We can obtain a directed graph for each generalised number by taking the generalised numbers as points and directing an edge from a generalised number toward each of its immediate predecessors.

It has been shown that these generalised numbers can be “added” and “multiplied” in a natural way by combining the associated graphs. The sum a+b is obtained by “hanging” the diagram of b onto that of a so the bottom point of a coincides with the top point of b. The product a·b is obtained by replacing each edge of the graph of a with the graph of b where the graphs are similarly oriented.

Paul Finsler, David Booth, Renatus Ziegler in Finsler set theory: platonism and circularity




image

  • I like Indonesian food better than Japanese food i ⪰ j, and
  • I like Japanese food better than English food j ⪰ e.
  • I also like French food better than English food f ⪰ e, but
  • I see French food as so different from the “exotic Eastern” foods that I can’t really say whether I prefer French food to Indonesian f≹i or Japanese f≹j.

    I would just be in a different mood if I wanted French food than if I wanted “exotic Eastern” food.

So my restaurant preferences are shaped like a poset. In a poset some things are comparable  and some things ain’t . Popularity is shaped as a poset and so is sexiness. Taste in movies is a poset too. The blood types have the same mathematical form as a poset but only if you reinterpret the relation  as “can donate to” rather than “is better than”. So not really the same as ethnic food.

image

Partial rankings | orders are transitive, so

  • (indonesian ⪰ japanese and japaneseenglish) implies indonesian ⪰ english.

That means I can use the “I prefer " symbol to codify what I said at the outset:

  • Indonesian  Japanese English
  • French English
  • neither⪰j nor⪰ f … nor⪰ f nor⪰ j (no comparison possible )


Posets correspond nicely to graphs since posets are multitrees.

image

image

Total orders — where any two things can be ranked  — also correspond to graphs, but the edges always line up the nodes into a one-dimensional path. So their graphs look less interesting and display less weird dimensional behaviour. Multitrees (posets) can have fractional numbers of dimensions, like 1.3 dimensions. That’s not really surprising since there are so many kinds of food / movies / attractiveness, and you probably haven’t spent the mental effort to precisely figure out what you think about how you rate all of them.

Rankings | orders are a nice way to say something mathematical without having to use traditional numbers.

I don’t need to score Indonesian at 95 and score Japanese at 85. Scores generated that way don’t mean as much as Zagat and US News & World Report would like you to think, anyway — certainly they don’t have all the properties that the numbers 85 and 95 have.

It’s more honest to just say Indonesian ⪰ Japanese lexicographically, and quantify no more.




At my secondary school, the high-scoring wide receiver was more popular than the fat lineman. And the fat lineman was more popular than the team statistician. But you couldn’t really compare the wide receiver’s popularity to that of the actor who got most of the lead roles. They were admired in different circles, to different degrees, by different people. With so little overlap, a hierarchy must treat them as separate rather than comparable.

So popularity = a partial order (and, possibly, an inverted arborescence or join-semilattice). Sometimes there is a binary relation  between two people such that one is-more-popular than the other. Sometimes you just can’t say. And no such relation exists. (neither geoff ≻ ian nor ian ≻ geoff)

Transitivity did hold at my school, so if you were more popular than geoff, you were by extension more popular than anyone than whom geoff was more popular. ( ari, shem, zvi: arishem and shemzvi implied arizvi)

And, by definition, even I was more popular than the nullset. (thanks, mathematics)




Blood types form a topological space (and a complete distributive lattice). There are three generators: A, B, and Rh+.

Above the “zero element” is the universal donor O− and the “unit element” is the universal receiver AB+.

A topological space contains a zero object, maybe other objects, and all unions & intersections  of anything in the space.  So taking the power set  of {A, B, +} yields the “power set topology” which I drew above. AB+ is the 1 object and “nullset” O− is the 0 object.

A lattice has joins  & meets  which function like  and  in a topological space. Like 1 or True in a Heyting algebra, blood type as a power-set topology has one “master” object AB+.




One example of a total ordering is the “hotness” scale from 110.

Because of the widespread disagreement about the meaning of the numbers, the only thing one can infer based on a man’s rating of a woman is that she is more attractive than those who score below her.

YOUTH

The Hotness Scale derives, I think, from a need to explain one’s tastes to peers and hear them justified.

It typically surfaces in sleepover conversations like this:

  • Chris (secretly likes Kelly Russell): Who do you think is hotter: Liz Jones, or Kelly Russell?
  • Dave: Are you kidding?! Liz Jones is waaaay hotter.
  • Chris: Oh, yeah. I mean, obviously. I was just checking. I just meant, you know, that I think Kelly Russell is like maybe a 7.
  • Dave: Are you crazy?! She’s like a 2.
  • Chris: Come on, 2 is like people who have skin grafts. 2 is people who were burned in fires.
  • Dave: Whatever. Maybe.
    (sheepish retreat to celebrity hotness)
    But man, Jane March is the hottest woman on the planet.
  • Chris: No way, Jenny McCarthy is hotter.

I’m a little embarrassed to admit (though I’ll still admit it) that when my best friend and I started using the hotness scale, we scored girls in different categories, like

  1. tan
  2. boobs
  3. personality
  4. legs
  5. I forget what else
  6. overall score

Yeah, we were really cool. (Also we were really twelve.)



DISPUTES

Fast forward to college. We guys were joking about using the ten-point scale, which by then was passé (although I did once use the phrase “a Bloomington 6 is a hometown 9”). We were trying to answer, what is the difference between a 6 and a 7 anyway? And is the distance between 6 and 7 greater or less than the distance between a 9 and a 10?

Everybody had taken calculus by this point so statements involving derivatives were bandied about (even though none of us meant to use real numbers … it was calculus as metaphor).

One guy proposed that each number should correspond to a decile —

  • 1 to the ugliest decile,
  • 10 to the hottest decile,
  • and so on.

 Someone else said that one’s initial reaction put a girl either in

  • the >4 (most of the time) or
  • <5 — but that since no one would ever hit on someone in the latter category,
  • in fact 1≈2≈3≈4.

Another said that he never assigned a 10 to anybody because that would mean he had met his wife. Um, yeah … we were still super cool.

Also contentious was whether each of us accepted the truth of whatever our own numerical ranking was. All I know is that whatever I said my score was, I secretly hoped it was 2 points higher.

There was a lot of inconsistency to the scores, which is why I’m bringing this up under the topic of rank without distance measure. Although I would wager that transitivity is violated, so perhaps this scale does not have a rational basis.



PEOPLE CAN BE ATTRACTIVE IN DIFFERENT WAYS

As I’m writing all of this I desperately want to jump ahead to partial orderings. But I haven’t defined them yet and I refuse to link to Wikipedia, so I’ll have to put that topic off.

Suffice to say that attraction is a perfect jumping-off point for one further generalization I want to make in order to get mathematics into bed with human experience.

Not everyone can be ranked side-by-side against everyone else. People can be attractive for different reasons (multiple >'s) and some people you just aren't comparable. All of these reasons and more are great justifications for switching to posets.



POSTSCRIPTA

I’d be interested to hear other interpretations of the hotness scale, or other scales of attractiveness, and how they evolved over time. What measure, if any, do you give guys when you’re in your 30’s or 40’s?

PS When I was 24, my girlfriend told me that “24 is a very hot age”. Ha ha.

PPS Tim Ferriss claims to be able to quantify the difference between a 6 and a 9. Tell that to Jimi Hendrix.