Why is math so ‘eerily effective’ in describing reality?
It might be very well possible that it isn't. I.e. the errors of our everyday language might not completely be weeded out by introducing clear symbols an axioms that rule their connections.
You're welcome (though I'm not sure what you're thanking me for). I'm not a mathematician and know very little about the Zeta function (I know what it is and that there is an unanswered question regarding it, but that's about it). From the abstract and conclusion, it seems related to Gödel's incompleteness theorems, and I'm more familiar with those.
As I understand it, Cantor in a sense predated Gödel with his diagonalization trick with the real numbers. Because the real numbers are problematic in various ways, I've long wondered if Kronecker had a point about God inventing the integers while the reals are from our imagination. I've long wondered if reality was rational but not real. And I suspect that reality doesn't have the precision implied by the reals.
That said, I'm still confronted with the eerily precise predictions made by the mathematics of both relativity and quantum mechanics. There are so many places where math is highly effective in describing reality. Maybe it's reality that has holes, which is why math does?
If with valid means 1>2 can be derived, then, either there's nothing wrong with reality but it is with the language with which we describe it. Or, perhaps, reality is like mathematical consistency all imagination.
Or, somewhere there's still an error in my paper; which has been reviewed for over more than a half year.
As I said, I'm not a mathematician, so I'm not qualified to review your paper, but if therein you do derive 2 < 1 — showing arithmetic to be inconsistent — then either you've established something shattering and ground-breaking or, yeah, there's a bug somewhere.
(I know that joke proofs that 1=2 or 1=0 usually rely on a hidden division by zero or other undefined operation that, once taken, invalidates the proof chain from that point on.)
As I said, I don't have the training to judge your paper. Rather than how long the paper has been available, I'd think the important metric is how many qualified mathematicians have reviewed and commented on it.
If you've really demonstrated that 2 < 1, I would think that's an extraordinary result, and the mathematical world should be beating a path to your door!
When it is published I will send you a copy. I think that, rightfully, people first would like to discuss/question the result. Th mathematicians are univerdity pro's up to and including the level of Professor. The publisher is Springer Nature.
I understand your hesitation. I would have it too!
Wouldn't this imply that maths is ultimately an empirical science? I don't think that's tenable, because it's just not how mathematicians math. It's not a process of abstraction or observation, but creation and (immaterial) discovery.
My own view follows Alfred North Whitehead in seeing 'eternal objects' (his version of platonic forms) as potentialities, available to be realised in many different ways. Then mathematical objects specifically would be potentialities for patterns/structures. We can explore the realm of possibilities simply by using our imaginations and playing with different ideas. And the actual world is necessarily possible, and therefore everything in it exists within the wider world of mere possibilities, explaining maths' effectiveness.
Depending on one's definition of "empirical", I think math is an empirical science, at least in terms of its discovery. The natural numbers come from our need to count and compare things like sheep or ships. The integers come from how subtraction doesn't always work with the natural numbers (8-5=3, but 5-8=?). The rational numbers come from our need to divide things and express ratios. The real numbers come from the ratio between a circle and its diameter or the √(2x²) result behind a square's diagonal. Even the complex numbers come from the need to make equations such as x²+1=0 have a valid answer.
So, yeah, I think in a very real sense the foundations of math come from our "experimenting" with numbers. Which all start with counting sheep and ships.
In your view, is there a difference between the imagined possibilities of unicorns and vampires versus the imagined possibilities of, say, relativity and quantum mechanics? Why are the latter so precise in predicting the outcomes of actual experiments whereas the former simply don't exist other than in our imaginations?
The natural numbers are pretty empirical, but beyond that I think it's too much of a stretch to call the others empirical. In a sense we can observe the number 2 in the real world (although even this is a stretch), but can we really observe a - 2? We can conceive of it and apply it, but I don't think I've ever seen - 2 of anything. The explanations you gave for the creation of the other kinds of numbers are not really empirical discovery so much as they're logical constructions and extensions of prior concepts.
The difference between imagined possibilities like unicorns vs imagined possibilities like relativity and QM is that the latter are both possible and actual.
An elevation below sea level might be an example of negative numbers. Owing someone money might be another. If the freezing and boiling points of water form a sensible scale, then we have negative temperatures. And I see rational numbers every time I order pizza. :)
But a lot of this depends on how you want to look at it. Countable sets seem sensible and physical to me. But the uncountable reals do seem problematic (in multiple ways). Even with countable sets, I think it's important to remember that "infinity" isn't a number, and things can get weird when it's used in math.
I've long liked the quote from Leopold Kronecker: "God made the integers, all else is the work of man."
It's a good quote! Although I think it overlooks the serious abstractness of the integers. Just like with negative numbers or rationals etc, it's all dependent on our act of deciding we're going to count things in a particular way with a particular frame of reference. Like we have to choose to measure elevation above/below sea level, and temperature with 100 as boiling and 0 as freezing, and pizza with the area of our pizza considered as our unit. We might equally say the pizza is 8 slices, rather than a slice is 1/8th of a pizza. And if we're counting sheep for example, is a lamb one or a fraction? Are conjoined twin sheep one, or two, or one and a half?
It's as though everything is numerable, but nothing is actually numbered until we get here and apply a frame of reference.
Indeed. That's kind of what this post is about — how we abstract from our experience. Nature doesn't do math.
For me, I think the difference between countable and uncountable has to do with what we can directly experience. Even the natural numbers are abstractions of the notion of cardinality. But for me the key is that we also directly experience the natural numbers. Likewise, the direct experience of, given a reasonable reference, "before" and "after" (or "above" and "below"). And before you cut the pizza, it's one unit that can be divided many ways. Other direct examples of division or ratios exist. Dividing 10 apples among three children or ratio of two adjacent sides of a rectangular field. We seem to encounter these things in the physical world.
We don't ever directly experience the real numbers, the smooth continuum. They pop out of our idealizations of circles and squares, but their physical reality seems super sus to me.
For the sheep, ha, I guess it would depend on how you viewed them. In terms of sale value, a lamb might have no value right now (but hopefully would down the line). If it's in terms of mouths to feed, every mouth would count!
Personally, I think the real numbers are about as physically real as any other numbers. If we just measure the diagonal of a square relative to its sides we find an irrational ratio. I suppose that's assuming Euclidean geometry which isn't strictly correct, at least universally, but I don't imagine relativity would dance around irratios any better.
But when we return to this slightly more grounded idea of real numbers as basically lengths, the "irrationality" of them isn't so strange. Who said that every length must be a multiple of a fraction of any other length? (IIRC it was the Pythagoreans) And why should we expect the number of hypothetically possible lengths of a piece of string to match the number of hypothetically possible collections of apples? There's really no reason to expect either to be true. I think the confusion comes from supposing they're basically the same sort of thing i.e. "numbers".
I think the confusion also comes from supposing that lines are made up of infinitely many dimensionless points, as if you could times 0 by infinity and get 1. But if we follow Aristotle and say that a line is not made up of points, but points on the line only present the potential for cutting the line (which also fits nicely with the real numbers as defined by Dedekind "cuts"), then we don't have an actual uncountable infinity of points in every line. We only have infinite potential for cutting the line, and even if we'd already made infinite cuts in it, there would still be opportunity for further cutting (because no amount of cutting could ever ever ever reduce a length to dimensionless points).
Kant indeed wrote about how our minds are structured to perceive reality in terms of time and space but he wrote much more important stuff on the subject of the realm of mathematical knowledge (what he would call synthetic a priori judgements) and our way of deducing our way into possible experience. In short: the nature of our cognition is such that it can only perceive things that are in line with these a priori judgements because every axiom of pure mathematics is already contained in the singular nature of consciousness itself; (and so are our other logical rules of cognition like for example causality, identity...). So in that sense mathematics isn't magically surprisingly effective in describing the world; it's the other way around. Things that don't align with these pure a priori judgement simply aren't object of possible experience. One could say Kant's system of a priory knowledge is based in the idea that the answer to the question of what something truly is (the real world) can never bypass the question of what knowledge is. From a Kantian perspective most systems before Kant that try to address reality are a case of putting the cart before the horse.
Your explanation that we perceive reality in terms of time and space because our brains are products of the physical world I would call already a departure from Kant (and maybe reflect a materialistic starting point?). Kant made complete abstraction from any physiological conditions for his deduction of the process of cognition.
Yes, there's no question I depart from Kant on several points (as I understand him — which might not be saying much). At the same time, what I said in the post in my mind goes towards explaining *why* what he said is true. There's also no question that my metaphysics is firmly rooted in physicalism. My starting point is the physical world.
Speaking of departures, I question the degree to which arithmetic and geometry really are synthetic. And as mentioned in the post the degree to which they are truly a priori. I plan to get into that more in the future. If I recall Kant's example (or maybe just a common example I've seen), the phrase "3+4=7" is synthetic because "7" isn't contained in "3+4". Yet in a sense it is. The equals-sign is more demanding than the verbal "is" — true identity is required. In what I think is a very real sense "3+4" (or "2+5" or infinite others) are all just different ways of "spelling" the cardinal number seven.
Or perhaps I just find the categorizing of "analytic" and "synthetic" restrictive and not useful. 🤷🏼♂️
As a kid, I did prefer the company of adults or people older than myself. Older people were a lot more interesting to me than my age peers. Part of the culture shock of my family moving from Minneapolis to Los Angeles mid-way through my seventh-grade year was indeed the very different "vibe" of California and all it entailed. At the time, I hated it. It took me three years to adjust and embrace the SoCal lifestyle. Which I ultimately did in a big way. By the time I was in high school, I was "a Californian". Here's a link to an old picture (from my other blog) of me in high school with my girlfriend:
Went through the same change when I transferred back to Minnesota. About three years of adjustment, but now I'm (again) a Minnesotan. Having lived in both places, I wouldn't move back. Way too many people there, and I no longer regard the California POV as highly. It's weird. I thrive on change, but not at first. Takes my mind a while to understand and adapt.
Funny how the ordinary farming everyone did is now "organic" farming. Both my parents came from rural backgrounds, though not farming or animal husbandry specifically. Mom's dad was a carpenter, and dad's dad was a Lutheran pastor. My dad did every year devote a good fraction of the backyard to a produce garden. And he had a compost heap and a worm farm ("red wigglers" if memory serves). He definitely was an "organic" grower.
I was a disappointment to my parents because I disliked most vegetables (except corn and potatoes). In my defense, my mom usually boiled the life out of them, and it wasn't until at least college when I tried steamed veggies that I found I could tolerate some of them. Still don't like most of them, though.
Sis and I mostly had a kind of détente as kids, but the older we got, the closer we got. We're best buds now, so there's always hope.
Sorry. I don't entirely get the connection between the barrels thing and the character of artists. It does raise the always interesting question about what connection one draws between the artist and their work. At the simplest level, the question of whether art should stand on its own or whether knowledge of the artist adds to or subtracts from the work. As an artist (not saying I'm a good one), I have a strong sense art should stand on its own, but I recognize that knowing the artist's background can inform the art. The question seems to become more urgent if one perceives the artist as a truly Bad Person (however one defines it). It seems one of those questions without a perfect answer.
Sounds like your days are busy! Mine are, too, but I'm inclined to plow more intellectual fields. I sometimes have to remind myself to go out for a walk. (The advantage to dog-sitting Bentley is that I'm committed to at least three daily walks with her.) But I'm typically head-down in whatever continuing self-education interests me plus whatever writing and coding I have on deck. Yesterday, I started transcribing my skydiving logbook for a post on my other blog. Brought back a lot of memories from that time.
Have fun in your garden! The mock apples are blooming here, which I love, so I plan to get out for a walk today and take some photos. They only bloom for a short while but are so pretty.
Sounds like you had a good weekend. Mine was mixed. My Twins are finally winning some games — eight in a row as of Sunday. The last six were home games, which seems their strength so far this season. They're 15-6 at home versus 6-14 away. On the season, they're 21-20 — that eight-game winning streak put them over .500 for the first time this year. They begin an away stretch today, so we'll see if they bring their success with them or leave it at home. 🤞🏼
OTOH, my smoke alarms have been erroring out — not false alarms, but error conditions indicating something wrong with the unit — installed last August but now three of the four are giving errors. Very annoying when it happens at 4:00 AM (three short but loud beeps every 60 seconds). I'll have to have the electrician out. Bonus, my garbage disposal has a leak, so I also need the plumber. It's always something. 😣
Sounds like you're definitely out in the boonies. I'm roughly 8 miles from downtown St. Paul and 14 from downtown Minneapolis (as the crow flies). There's mostly farmland to the south, though. Quite some time ago, I lived in a townhouse about 15 miles south of the Twin Cities. Was half suburbia and half farmland. Lots of corn fields, but while I lived there, I saw several of them turned into housing developments. Our ever-expanding population. The thing about growth curves is that they can't go on indefinitely. Something has to give, and it won't be pretty when it does. COVID killed over a million of us, and it's probably a matter of time before there's another epidemic of some kind.
The really sad thing about COVID is that our death rate was 0.350%. Compare that to the other two similarly sized countries: India (0.038%) and China (0.009%). We ranked #17 in terms of %-of-deaths compared to their #159 and #201 ranks, respectively. With the current administration, another epidemic seems certain to be worse for us. This is what happens when idiots rule.
I'm a bit over 8 miles from the international airport here and slightly off the usual flight paths, so the planes don't go overhead, but I see lots of them coming in for a landing or, when the wind shifts, taking off. I've always loved aviation, so I enjoy seeing them. There's a small airport just east of me, so I see small planes frequently. For about a year, I was into skydiving, so I've experienced a lot more take offs than landings. 😁
I've decided to get busy on some home projects, so I may not be around as much as usual. And I have to deal with the smoke alarms and garbage disposal, so may only check in here from time to time.
Yep, Twins won both games of the double-header and became the first MLB team to win 10 in a row this season. Nice to see them win a couple on the road. If they beat the O's again today, they'll sweep the series.
Pandemics are just one example of how growth curves can get stymied. All growth curves, especially exponential ones, are necessarily self-limiting. Nothing can grow forever. An animal population can out-grow its food supply, for instance. In the case of human disease, population density and our ability to travel become key factors.
Heh. I'll take the planes; you can take the coffee. I used to love flying. It's too much of a PITA nowadays, and I've had my fill. There was a period when I was working that my company flew me around a lot to do installs and training of a device that I was their only specialist for. I visited a lot of major telco switching centers and a handful of atomic plants. And the Pentagon. Great stuff for a tech geek!
I don't like hot beverages in general, and coffee always tastes like dirty brown water to me. I went through a phase of drinking hot tea, but it didn't last. The one exception is hot cider with cinnamon in winter, but even that's rare. Oddly, I like coffee candy and coffee ice cream.
It's not that I actually considered being a standup comedian, but that I realized there could easily be an alternate world where I became one. I've long loved watching standup comedy, and I can imagine doing it, but that's a fairly recent thought. I was more interested in making movies or TV shows.
There's a saying that you don't own a home, the home owns you. That seems more true than not to me. My project (other than the damn smoke alarms) is basically just spring cleaning and aggressive tossing of stuff that, at this point in my life, is just collecting dust. I have far too many books I'll never read again, and DVDs I'll never watch again, and a ton of music CDs that I long ago transferred to my PC and phone so will never play again. I've already donated a lot of hardcover books to the library, and I need to figure out how to give all this other stuff a good home. I hate the idea of just tossing them, but given streaming these days, their value is close to nil.
Have a good summer! We'll chat when we can. Sounds like we both have other things pulling our attention. The ebb and flow of life!
I suppose it's all down to priorities and interests. BentleyMom does some gardening in her backyard — flowers and some veggies and herbs — and likes working on her yard whereas I live in a condo in part so I don't have to even cut the grass. In younger days I enjoyed camping, and I still love going for a walk when my back permits it, but gardening (let alone farming) not so much.
Chicken, potatoes, and corn sounds yummy. I don't cook unless heating things up in the microwave or toaster oven counts. I sometimes boil up some pasta. At home on my own I tend to view food as just fuel. Like a dog, I'm fine eating the same things every day.
I think, at least in my mom's case, boiling veggies was all she knew. Both mom and dad were intellectuals with rich inner lives but not much attachment to the physical world. We could never afford to be fashionable or trendy and didn't much care to be. Great parents but mom wasn't much of a cook. She did make some excellent breads, though.
I'll overlook a lot in the personal character of an artist (though I do have my limits). I'll take what's good from the work. Agatha Christie is an example. She was a bit of a racist and Imperialist but also definitely a product of her time. I can shrug that off because she really was one of the best mystery writers ever and, for her time, was more open-minded than most.
No, I only have Bentley when BentleyMom needs a break or goes for a vacation. And since COVID she works from home a fair bit. Before that I used to go walk Bentley once in a while, but not so much since.
I'm not sure what you mean by "it" as the only path. Or by "defining or discovering reality". If you mean math, then no. Math is often powerful in describing reality, so it often has useful application. It's not uncommon for a mathematician to find some abstract mathematics that they're certain will never have real-life application, but down the road it turns out to have. For example, the encryption schemes we depend on in so many sectors started out as some interesting, but "useless", math. I would say we discover reality through the sum of our experiences, and I'm not sure it can be defined.
No specific back injury (though in my field tech days I did some heavy lifting). As far as I know, it's just aging. Possibly exacerbated by a tendency to slouch when sitting.
There's a library nearby, and a market. Visiting bookstores require a car. So do the library and market, but they are theoretically within walking distance. When I was able to do 4- or 5-mile walks, I sometimes walked past them. I don't drink coffee, so don't know much about coffee shops. There's a Starbucks in the area. No bistro-type places I know of, other than the Barnes & Noble has a coffee shop inside. Downtowns St. Paul and Minneapolis surely have them. How about you? I get the impression you're "out in the boonies" so to speak?
I've read blog posts by single people who get into cooking for themselves, but as with you, most of the single people I know don't spend much energy on it. I enjoy restaurants, especially ones I've never been to. I know it wouldn't be to your taste, but BentleyMom and I are on a slow-paced mission to explore craft burger and craft beer places. We both enjoy both. Do you eat out much?
My "Math Musings" newsletter is about math, and I've written about it a fair bit on my old blog. Not that I'm in any way a mathematician. My math skills only take me as far as basic calculus derivatives and even more basic integration. But it does interest me, and a while back I started trying to teach myself the math necessary to manage quantum mechanics (because I kept hearing that no words really convey it — one has to learn the math, and that turned out to be correct, but I'm still chewing on it). My "Quantum Curious" newsletter features some of my explorations (again, there's much more on my old blog — I haven't been inclined to repeat too many of those posts here but some).
"Bioregional Based Eating and Economies" does seem self-explanatory!
Have a great weekend. We're not expecting rain until late next week, but clouds rolling in during the last hour or so sure seem to threaten it.
Thank you. You asked:
Why is math so ‘eerily effective’ in describing reality?
It might be very well possible that it isn't. I.e. the errors of our everyday language might not completely be weeded out by introducing clear symbols an axioms that rule their connections.
Viz.
https://osf.io/preprints/osf/ceqdy_v2
This paper is accepted for publication. Meaning it is in the press.
You're welcome (though I'm not sure what you're thanking me for). I'm not a mathematician and know very little about the Zeta function (I know what it is and that there is an unanswered question regarding it, but that's about it). From the abstract and conclusion, it seems related to Gödel's incompleteness theorems, and I'm more familiar with those.
As I understand it, Cantor in a sense predated Gödel with his diagonalization trick with the real numbers. Because the real numbers are problematic in various ways, I've long wondered if Kronecker had a point about God inventing the integers while the reals are from our imagination. I've long wondered if reality was rational but not real. And I suspect that reality doesn't have the precision implied by the reals.
That said, I'm still confronted with the eerily precise predictions made by the mathematics of both relativity and quantum mechanics. There are so many places where math is highly effective in describing reality. Maybe it's reality that has holes, which is why math does?
Thank you (for article and reply).
If with valid means 1>2 can be derived, then, either there's nothing wrong with reality but it is with the language with which we describe it. Or, perhaps, reality is like mathematical consistency all imagination.
Or, somewhere there's still an error in my paper; which has been reviewed for over more than a half year.
As I said, I'm not a mathematician, so I'm not qualified to review your paper, but if therein you do derive 2 < 1 — showing arithmetic to be inconsistent — then either you've established something shattering and ground-breaking or, yeah, there's a bug somewhere.
(I know that joke proofs that 1=2 or 1=0 usually rely on a hidden division by zero or other undefined operation that, once taken, invalidates the proof chain from that point on.)
Okay.. but note the 1.3 year review.
When the paper "comes out of the press", I will send you. Moreover, there are no divisions zero here. Only, l'Hôpital
Sorry, 0.5 year review.
Let me put it like so.
Where do you see that my argument in
https://osf.io/preprints/osf/ceqdy_v2
is based on 1=0? Not much deep mathematics needed to find it, if it's there.
As I said, I don't have the training to judge your paper. Rather than how long the paper has been available, I'd think the important metric is how many qualified mathematicians have reviewed and commented on it.
If you've really demonstrated that 2 < 1, I would think that's an extraordinary result, and the mathematical world should be beating a path to your door!
When it is published I will send you a copy. I think that, rightfully, people first would like to discuss/question the result. Th mathematicians are univerdity pro's up to and including the level of Professor. The publisher is Springer Nature.
I understand your hesitation. I would have it too!
To be Published in: Lobachevski Journal of Mathematics, No3 2025.
There's (a part of) your metric.
Wouldn't this imply that maths is ultimately an empirical science? I don't think that's tenable, because it's just not how mathematicians math. It's not a process of abstraction or observation, but creation and (immaterial) discovery.
My own view follows Alfred North Whitehead in seeing 'eternal objects' (his version of platonic forms) as potentialities, available to be realised in many different ways. Then mathematical objects specifically would be potentialities for patterns/structures. We can explore the realm of possibilities simply by using our imaginations and playing with different ideas. And the actual world is necessarily possible, and therefore everything in it exists within the wider world of mere possibilities, explaining maths' effectiveness.
Depending on one's definition of "empirical", I think math is an empirical science, at least in terms of its discovery. The natural numbers come from our need to count and compare things like sheep or ships. The integers come from how subtraction doesn't always work with the natural numbers (8-5=3, but 5-8=?). The rational numbers come from our need to divide things and express ratios. The real numbers come from the ratio between a circle and its diameter or the √(2x²) result behind a square's diagonal. Even the complex numbers come from the need to make equations such as x²+1=0 have a valid answer.
So, yeah, I think in a very real sense the foundations of math come from our "experimenting" with numbers. Which all start with counting sheep and ships.
In your view, is there a difference between the imagined possibilities of unicorns and vampires versus the imagined possibilities of, say, relativity and quantum mechanics? Why are the latter so precise in predicting the outcomes of actual experiments whereas the former simply don't exist other than in our imaginations?
The natural numbers are pretty empirical, but beyond that I think it's too much of a stretch to call the others empirical. In a sense we can observe the number 2 in the real world (although even this is a stretch), but can we really observe a - 2? We can conceive of it and apply it, but I don't think I've ever seen - 2 of anything. The explanations you gave for the creation of the other kinds of numbers are not really empirical discovery so much as they're logical constructions and extensions of prior concepts.
The difference between imagined possibilities like unicorns vs imagined possibilities like relativity and QM is that the latter are both possible and actual.
An elevation below sea level might be an example of negative numbers. Owing someone money might be another. If the freezing and boiling points of water form a sensible scale, then we have negative temperatures. And I see rational numbers every time I order pizza. :)
But a lot of this depends on how you want to look at it. Countable sets seem sensible and physical to me. But the uncountable reals do seem problematic (in multiple ways). Even with countable sets, I think it's important to remember that "infinity" isn't a number, and things can get weird when it's used in math.
I've long liked the quote from Leopold Kronecker: "God made the integers, all else is the work of man."
It's a good quote! Although I think it overlooks the serious abstractness of the integers. Just like with negative numbers or rationals etc, it's all dependent on our act of deciding we're going to count things in a particular way with a particular frame of reference. Like we have to choose to measure elevation above/below sea level, and temperature with 100 as boiling and 0 as freezing, and pizza with the area of our pizza considered as our unit. We might equally say the pizza is 8 slices, rather than a slice is 1/8th of a pizza. And if we're counting sheep for example, is a lamb one or a fraction? Are conjoined twin sheep one, or two, or one and a half?
It's as though everything is numerable, but nothing is actually numbered until we get here and apply a frame of reference.
Indeed. That's kind of what this post is about — how we abstract from our experience. Nature doesn't do math.
For me, I think the difference between countable and uncountable has to do with what we can directly experience. Even the natural numbers are abstractions of the notion of cardinality. But for me the key is that we also directly experience the natural numbers. Likewise, the direct experience of, given a reasonable reference, "before" and "after" (or "above" and "below"). And before you cut the pizza, it's one unit that can be divided many ways. Other direct examples of division or ratios exist. Dividing 10 apples among three children or ratio of two adjacent sides of a rectangular field. We seem to encounter these things in the physical world.
We don't ever directly experience the real numbers, the smooth continuum. They pop out of our idealizations of circles and squares, but their physical reality seems super sus to me.
For the sheep, ha, I guess it would depend on how you viewed them. In terms of sale value, a lamb might have no value right now (but hopefully would down the line). If it's in terms of mouths to feed, every mouth would count!
Personally, I think the real numbers are about as physically real as any other numbers. If we just measure the diagonal of a square relative to its sides we find an irrational ratio. I suppose that's assuming Euclidean geometry which isn't strictly correct, at least universally, but I don't imagine relativity would dance around irratios any better.
But when we return to this slightly more grounded idea of real numbers as basically lengths, the "irrationality" of them isn't so strange. Who said that every length must be a multiple of a fraction of any other length? (IIRC it was the Pythagoreans) And why should we expect the number of hypothetically possible lengths of a piece of string to match the number of hypothetically possible collections of apples? There's really no reason to expect either to be true. I think the confusion comes from supposing they're basically the same sort of thing i.e. "numbers".
I think the confusion also comes from supposing that lines are made up of infinitely many dimensionless points, as if you could times 0 by infinity and get 1. But if we follow Aristotle and say that a line is not made up of points, but points on the line only present the potential for cutting the line (which also fits nicely with the real numbers as defined by Dedekind "cuts"), then we don't have an actual uncountable infinity of points in every line. We only have infinite potential for cutting the line, and even if we'd already made infinite cuts in it, there would still be opportunity for further cutting (because no amount of cutting could ever ever ever reduce a length to dimensionless points).
this is the good stuff!!
Kant indeed wrote about how our minds are structured to perceive reality in terms of time and space but he wrote much more important stuff on the subject of the realm of mathematical knowledge (what he would call synthetic a priori judgements) and our way of deducing our way into possible experience. In short: the nature of our cognition is such that it can only perceive things that are in line with these a priori judgements because every axiom of pure mathematics is already contained in the singular nature of consciousness itself; (and so are our other logical rules of cognition like for example causality, identity...). So in that sense mathematics isn't magically surprisingly effective in describing the world; it's the other way around. Things that don't align with these pure a priori judgement simply aren't object of possible experience. One could say Kant's system of a priory knowledge is based in the idea that the answer to the question of what something truly is (the real world) can never bypass the question of what knowledge is. From a Kantian perspective most systems before Kant that try to address reality are a case of putting the cart before the horse.
Your explanation that we perceive reality in terms of time and space because our brains are products of the physical world I would call already a departure from Kant (and maybe reflect a materialistic starting point?). Kant made complete abstraction from any physiological conditions for his deduction of the process of cognition.
Thank you, I'm glad you enjoyed it!
Yes, there's no question I depart from Kant on several points (as I understand him — which might not be saying much). At the same time, what I said in the post in my mind goes towards explaining *why* what he said is true. There's also no question that my metaphysics is firmly rooted in physicalism. My starting point is the physical world.
Speaking of departures, I question the degree to which arithmetic and geometry really are synthetic. And as mentioned in the post the degree to which they are truly a priori. I plan to get into that more in the future. If I recall Kant's example (or maybe just a common example I've seen), the phrase "3+4=7" is synthetic because "7" isn't contained in "3+4". Yet in a sense it is. The equals-sign is more demanding than the verbal "is" — true identity is required. In what I think is a very real sense "3+4" (or "2+5" or infinite others) are all just different ways of "spelling" the cardinal number seven.
Or perhaps I just find the categorizing of "analytic" and "synthetic" restrictive and not useful. 🤷🏼♂️
🙋🏼♂️ Yo, JT, I'm over here!
Continuing the conversation from this thread:
https://logosconcarne.substack.com/p/unblocking-the-universe/comment/115079013
As a kid, I did prefer the company of adults or people older than myself. Older people were a lot more interesting to me than my age peers. Part of the culture shock of my family moving from Minneapolis to Los Angeles mid-way through my seventh-grade year was indeed the very different "vibe" of California and all it entailed. At the time, I hated it. It took me three years to adjust and embrace the SoCal lifestyle. Which I ultimately did in a big way. By the time I was in high school, I was "a Californian". Here's a link to an old picture (from my other blog) of me in high school with my girlfriend:
https://logosconcarne.wordpress.com/wp-content/uploads/2012/08/gf-sps.png
Went through the same change when I transferred back to Minnesota. About three years of adjustment, but now I'm (again) a Minnesotan. Having lived in both places, I wouldn't move back. Way too many people there, and I no longer regard the California POV as highly. It's weird. I thrive on change, but not at first. Takes my mind a while to understand and adapt.
Funny how the ordinary farming everyone did is now "organic" farming. Both my parents came from rural backgrounds, though not farming or animal husbandry specifically. Mom's dad was a carpenter, and dad's dad was a Lutheran pastor. My dad did every year devote a good fraction of the backyard to a produce garden. And he had a compost heap and a worm farm ("red wigglers" if memory serves). He definitely was an "organic" grower.
I was a disappointment to my parents because I disliked most vegetables (except corn and potatoes). In my defense, my mom usually boiled the life out of them, and it wasn't until at least college when I tried steamed veggies that I found I could tolerate some of them. Still don't like most of them, though.
Sis and I mostly had a kind of détente as kids, but the older we got, the closer we got. We're best buds now, so there's always hope.
Sorry. I don't entirely get the connection between the barrels thing and the character of artists. It does raise the always interesting question about what connection one draws between the artist and their work. At the simplest level, the question of whether art should stand on its own or whether knowledge of the artist adds to or subtracts from the work. As an artist (not saying I'm a good one), I have a strong sense art should stand on its own, but I recognize that knowing the artist's background can inform the art. The question seems to become more urgent if one perceives the artist as a truly Bad Person (however one defines it). It seems one of those questions without a perfect answer.
Sounds like your days are busy! Mine are, too, but I'm inclined to plow more intellectual fields. I sometimes have to remind myself to go out for a walk. (The advantage to dog-sitting Bentley is that I'm committed to at least three daily walks with her.) But I'm typically head-down in whatever continuing self-education interests me plus whatever writing and coding I have on deck. Yesterday, I started transcribing my skydiving logbook for a post on my other blog. Brought back a lot of memories from that time.
Have fun in your garden! The mock apples are blooming here, which I love, so I plan to get out for a walk today and take some photos. They only bloom for a short while but are so pretty.
Sounds like you had a good weekend. Mine was mixed. My Twins are finally winning some games — eight in a row as of Sunday. The last six were home games, which seems their strength so far this season. They're 15-6 at home versus 6-14 away. On the season, they're 21-20 — that eight-game winning streak put them over .500 for the first time this year. They begin an away stretch today, so we'll see if they bring their success with them or leave it at home. 🤞🏼
OTOH, my smoke alarms have been erroring out — not false alarms, but error conditions indicating something wrong with the unit — installed last August but now three of the four are giving errors. Very annoying when it happens at 4:00 AM (three short but loud beeps every 60 seconds). I'll have to have the electrician out. Bonus, my garbage disposal has a leak, so I also need the plumber. It's always something. 😣
Sounds like you're definitely out in the boonies. I'm roughly 8 miles from downtown St. Paul and 14 from downtown Minneapolis (as the crow flies). There's mostly farmland to the south, though. Quite some time ago, I lived in a townhouse about 15 miles south of the Twin Cities. Was half suburbia and half farmland. Lots of corn fields, but while I lived there, I saw several of them turned into housing developments. Our ever-expanding population. The thing about growth curves is that they can't go on indefinitely. Something has to give, and it won't be pretty when it does. COVID killed over a million of us, and it's probably a matter of time before there's another epidemic of some kind.
The really sad thing about COVID is that our death rate was 0.350%. Compare that to the other two similarly sized countries: India (0.038%) and China (0.009%). We ranked #17 in terms of %-of-deaths compared to their #159 and #201 ranks, respectively. With the current administration, another epidemic seems certain to be worse for us. This is what happens when idiots rule.
I'm a bit over 8 miles from the international airport here and slightly off the usual flight paths, so the planes don't go overhead, but I see lots of them coming in for a landing or, when the wind shifts, taking off. I've always loved aviation, so I enjoy seeing them. There's a small airport just east of me, so I see small planes frequently. For about a year, I was into skydiving, so I've experienced a lot more take offs than landings. 😁
I've decided to get busy on some home projects, so I may not be around as much as usual. And I have to deal with the smoke alarms and garbage disposal, so may only check in here from time to time.
Have fun farming!
Yep, Twins won both games of the double-header and became the first MLB team to win 10 in a row this season. Nice to see them win a couple on the road. If they beat the O's again today, they'll sweep the series.
Pandemics are just one example of how growth curves can get stymied. All growth curves, especially exponential ones, are necessarily self-limiting. Nothing can grow forever. An animal population can out-grow its food supply, for instance. In the case of human disease, population density and our ability to travel become key factors.
Heh. I'll take the planes; you can take the coffee. I used to love flying. It's too much of a PITA nowadays, and I've had my fill. There was a period when I was working that my company flew me around a lot to do installs and training of a device that I was their only specialist for. I visited a lot of major telco switching centers and a handful of atomic plants. And the Pentagon. Great stuff for a tech geek!
I don't like hot beverages in general, and coffee always tastes like dirty brown water to me. I went through a phase of drinking hot tea, but it didn't last. The one exception is hot cider with cinnamon in winter, but even that's rare. Oddly, I like coffee candy and coffee ice cream.
It's not that I actually considered being a standup comedian, but that I realized there could easily be an alternate world where I became one. I've long loved watching standup comedy, and I can imagine doing it, but that's a fairly recent thought. I was more interested in making movies or TV shows.
There's a saying that you don't own a home, the home owns you. That seems more true than not to me. My project (other than the damn smoke alarms) is basically just spring cleaning and aggressive tossing of stuff that, at this point in my life, is just collecting dust. I have far too many books I'll never read again, and DVDs I'll never watch again, and a ton of music CDs that I long ago transferred to my PC and phone so will never play again. I've already donated a lot of hardcover books to the library, and I need to figure out how to give all this other stuff a good home. I hate the idea of just tossing them, but given streaming these days, their value is close to nil.
Have a good summer! We'll chat when we can. Sounds like we both have other things pulling our attention. The ebb and flow of life!
FWIW, Substack lets you edit your own comments. Click or press the "..." icon that goes with the comment and select "Edit".
I suppose it's all down to priorities and interests. BentleyMom does some gardening in her backyard — flowers and some veggies and herbs — and likes working on her yard whereas I live in a condo in part so I don't have to even cut the grass. In younger days I enjoyed camping, and I still love going for a walk when my back permits it, but gardening (let alone farming) not so much.
Chicken, potatoes, and corn sounds yummy. I don't cook unless heating things up in the microwave or toaster oven counts. I sometimes boil up some pasta. At home on my own I tend to view food as just fuel. Like a dog, I'm fine eating the same things every day.
I think, at least in my mom's case, boiling veggies was all she knew. Both mom and dad were intellectuals with rich inner lives but not much attachment to the physical world. We could never afford to be fashionable or trendy and didn't much care to be. Great parents but mom wasn't much of a cook. She did make some excellent breads, though.
I'll overlook a lot in the personal character of an artist (though I do have my limits). I'll take what's good from the work. Agatha Christie is an example. She was a bit of a racist and Imperialist but also definitely a product of her time. I can shrug that off because she really was one of the best mystery writers ever and, for her time, was more open-minded than most.
No, I only have Bentley when BentleyMom needs a break or goes for a vacation. And since COVID she works from home a fair bit. Before that I used to go walk Bentley once in a while, but not so much since.
I'm not sure what you mean by "it" as the only path. Or by "defining or discovering reality". If you mean math, then no. Math is often powerful in describing reality, so it often has useful application. It's not uncommon for a mathematician to find some abstract mathematics that they're certain will never have real-life application, but down the road it turns out to have. For example, the encryption schemes we depend on in so many sectors started out as some interesting, but "useless", math. I would say we discover reality through the sum of our experiences, and I'm not sure it can be defined.
Enjoy your gardening!
No specific back injury (though in my field tech days I did some heavy lifting). As far as I know, it's just aging. Possibly exacerbated by a tendency to slouch when sitting.
There's a library nearby, and a market. Visiting bookstores require a car. So do the library and market, but they are theoretically within walking distance. When I was able to do 4- or 5-mile walks, I sometimes walked past them. I don't drink coffee, so don't know much about coffee shops. There's a Starbucks in the area. No bistro-type places I know of, other than the Barnes & Noble has a coffee shop inside. Downtowns St. Paul and Minneapolis surely have them. How about you? I get the impression you're "out in the boonies" so to speak?
I've read blog posts by single people who get into cooking for themselves, but as with you, most of the single people I know don't spend much energy on it. I enjoy restaurants, especially ones I've never been to. I know it wouldn't be to your taste, but BentleyMom and I are on a slow-paced mission to explore craft burger and craft beer places. We both enjoy both. Do you eat out much?
My "Math Musings" newsletter is about math, and I've written about it a fair bit on my old blog. Not that I'm in any way a mathematician. My math skills only take me as far as basic calculus derivatives and even more basic integration. But it does interest me, and a while back I started trying to teach myself the math necessary to manage quantum mechanics (because I kept hearing that no words really convey it — one has to learn the math, and that turned out to be correct, but I'm still chewing on it). My "Quantum Curious" newsletter features some of my explorations (again, there's much more on my old blog — I haven't been inclined to repeat too many of those posts here but some).
"Bioregional Based Eating and Economies" does seem self-explanatory!
Have a great weekend. We're not expecting rain until late next week, but clouds rolling in during the last hour or so sure seem to threaten it.