In an earlier post I talked a bit about complex numbers and the great, great, mathematician Augustin-Louis Cauchy (1789 - 1857), who pretty much single-handedly invented the discipline of complex analysis (he was very prolific and did lots of other great stuff too).
Well, that gave me terrible flashbacks to 1st year Pure Maths at uni (way back in the Cretaceous Period). Invent i to be the square root of minus one. As for lemmas where you assume what you want to prove and then prove it by assuming the proof (I think that is how those things went but it was so long ago. I fled Maths after that and stuck to English Lit and Psychology and Philosophy. A weird species, mathematicians!
You're probably talking about a proof method called "proof by contradiction" (there's a fancy latin name: modus bonehead, or something like that). You *assume* something is true, then work out the consequences of that assumption being true. If these new consequences you derive are inconsistent (eg they lead to something like 1 = 2) then it means your original assumption is false.
An example of the kind of assumption made might be "assume any odd number can be factored by 2" - you could then show that this assumption leads to a nonsensical result and so the original assumption must be wrong: you've then shown that an odd number can't be (perfectly) divided by 2. This is a bit of a silly example to get the idea across.
Choosing the qualifier is important - you might want to phrase it better as "assume there exists one odd number that can be perfectly divided by 2" and then show that this assumption leads to a contradiction, thereby proving that there exists no such odd number.
Yes, that sounds more like it. Trouble is, as I get older, each new bit of knowledge that gets into my brain pushes out an older bit of knowledge, so my memory of Uni Maths all those decades ago gets seriously degraded.
The reason the budding scholarship in the islamic world petered out was because it came into conflict with the priesthood, to put it bluntly.
To the priests, even a thousand years ago, the Quran contained all necessary knowledge since it was the word of god by way of the angel who dictated to the prophet who, being illiterate, repeated it to a scribe.
(The kadis and imams of the time had obviously never heard of 'chinese whispers'.)
This set-in-stone approach was very popular with the secular leaders of the islamic world, and mainly being arabs or influenced by arabs and therefore lacking any scientific or even civilisational basis (the arabs' achievements were mainly repossessed greco-roman-egyptian ones in new packaging, and being smart enough to use clever conquered peoples wok to the benefit of the state) above tribal level, they opted for the priests' line of thinking, leading to islamic scholars either renouncing their idas, getting murdered or moving to China or Europe.
(There's a lesson in there for parlour-pinks, wokesters and sundry.)
The above is also the reason why Europe started surpassing the islamic world after the Black Plague had done its job: here, new ideas were welcome as long as they didn't challenge papal dogma, and the catholic church virtually killed its own moral authority by its - even at the time! - well-known depravity, until Luther popped the zit called Pope. Had the islamic world produced a similar figure as Martin Luther... but islamic-arabic culture is incapable of that it seems, being firmly rooted in clan-structures and inbreeding (yes, marrying first cousins for ten centuries is inbreeding, it's not racism but genetics and genetics doesn't care about feelings).
Again, there's a lesson: don't muck about with anything that has to do with fertility, hormones and heritability.
Speaking of exponential growth: a wild sow can have 2-3 litters per year, of 6-8 "boarlets". Without predators, almost all of them will reach maturity inside a year and start reproducing.
Q: Starting with 100 females and 100 males (Yes, Virginia, there are only two sexes) and assuming no more than 10% of "boarlets" die before reproducing, and that all survivors reach at least 5 years of age; when do the population of boar reach 100 000 adult animals?
The above is too difficult for 100% of all feminists, LGBTP, Greens, woke, liberals, progressives and modern-day po-mo marxists I've asked.
Which is the reason Sweden have a very real problem with far more than 100 000 boars destroying crops and gardens.
Yes - that's my speculation too. I suspect that the religious 'scholars' were more influential than the 'academic' scholars.
We've sort of witnessed something a bit similar with regards to covid. You can see it "between the lines", so to speak, where some authors published some result that was clearly against the 'narrative' when you examined it carefully, but the abstract and conclusions were written so as to appear to *support* the narrative.
We have it in the GenderWoo field where it's next to impossible to get 'honest' research funded.
Even the prestigious journal Nature ran an editorial basically stating that some research should be rejected on *moral* grounds (i.e. if it supported a conclusion or looked to be investigating something that might upset the Woke wagon).
Uh-oh. (Pause to get another cup of coffee and reread.)
Ok, now that I’m caffeinated, I’ve decided I am enjoying your “complex” series. Human intellect need not be cowed by the boundaries of what first appears rational or possible. (Even if I backed slowly away from some of the math you presented like it was a transwoman in a thong.) Instead of reflexively mocking crazy-seeming ideas, we can test them— and that gives us the square root of negative 1. (But not “oh, look, that transwoman in a thong is pregnant!”) We want to do this; there is something surprising and indeed delightful about being able to. But when there is no sense to be found, only nonsense, well… back to mocking, one hopes. (Unless one is the transwoman with a beer gut. Who is brave and beautiful and I hope doesn’t come after me because she is a foot taller than me and a hundred pounds heavier.)
Euler said it best. e^i*pi = -1. an incredible unification of the complex, exponential, unitary, geometric, negative and if rewritten, zero. Greatest equation of all time?
Rudolph: Scholarship in the Muslim world around the time of al-Haytham was much more advanced than anything that existed in Europe at the time, but it kind of fizzled out and never really went anywhere. I don’t know why.
Good question. For a bit of a long winded answer, though still a fascinating one, see this oldish essay in "The New Atlantis" on "Why the Arabic World Turned Away from Science". As another commenter suggested or argued, part of the reason was that science conflicted with -- STILL conflicts with --religious dogma. Though there are still some other interesting aspects:
"There is a final reason why it makes little sense to exhort Muslims to their own past: while there are many things that the Islamic world lacks, pride in heritage is not one of them. What is needed in Islam is less self-pride and more self-criticism. Today, self-criticism in Islam is valued only insofar as it is made as an appeal to be more pious and less spiritually corrupt. And yet most criticism in the Muslim world is directed outward, at the West. This prejudice — what Fouad Ajami has called (referring to the Arab world) 'a political tradition of belligerent self-pity' — is undoubtedly one of Islam’s biggest obstacles. It makes information that contradicts orthodox belief irrelevant, and it closes off debate about the nature and history of Islam."
Thank you for digging that one up, it's a really good long-form article with great references!
"But the Islamic turn away from scholarship actually preceded the civilization’s geopolitical decline — it can be traced back to the rise of the anti-philosophical Ash’arism school among Sunni Muslims, who comprise the vast majority of the Muslim world."
This quote really sums it up, and much better than my rambling atempt earlier.
"While the Mu’tazilites had contended that the Koran was created and so God’s purpose for man must be interpreted through reason, the Ash’arites believed the Koran to be coeval with God — and therefore unchallengeable. At the heart of Ash’ari metaphysics is the idea of occasionalism, a doctrine that denies natural causality. Put simply, it suggests natural necessity cannot exist because God’s will is completely free. Ash’arites believed that God is the only cause, so that the world is a series of discrete physical events each willed by God."
This is how virtually all moslems reason today, which makes it completely impossible to "integrate" them or for them to integrate with any other society or civilisation; whatever scientific qeury you have, in any filed of scence, the answer is always "God wills it" anyway.
The number of times I've heard and seen arab children in Sweden, iraqis and palestinians especially, answer with a shrug and a "Because God" to any question relating to science is in three digits.
"Why does the sky change colour at dawn and at evening?"
"Bcause God wants it to."
Completely serious, no sarcasm.
Small wonder the Woke are in love with islam (or rather their fantasy-version of it.)
De nada; share the wealth; praise the lord & pass the ammunition ... 🙂. I've gotten a fair amount of "mileage" out of it so I like to pass it around whenever I get the chance.
Rikard: "... completely impossible to 'integrate' them or for them to integrate with any other society or civilization ..."
Amen to that. Muslims generally don't play well with anyone -- as I think many countries, Sweden in particular, are finding to their "chagrin" or misfortune. But somewhat apropos of which, see my 2018 archived article in Canada's Post Millennial asking, "Can Islam, Sharia and a Secular Democracy coincide?" -- though I had titled my submission to them using "coexist" instead of "coincide":
Of particular note therein is a quote of Muslim reformer Shireen Qudosi who said, in an article of hers at The Blaze, "either Islam needs to evolve or it needs to die" -- don't see much evidence of the former.
But for another bit of Canadiana on the Muslim front -- which I apparently wasn't aware of at the time I wrote that article, see this quote of "Salim Mansur, professor of political science at the University of Western Ontario" in his rather damning testimony to Canada's "Standing Committee On Citizenship And Immigration" in 2012:
Mansur: "We have the precedent of how we selectively closed immigration from the Soviet bloc countries during the Cold War years, and we need to consider doing the same in terms of immigration from Muslim countries for a period of time given how disruptive is the cultural baggage of illiberal values that is brought in as a result. ....
Lest any member wants to instruct me that my views are in any way politically incorrect, or worse, I would like members to note that I come before you as a practising Muslim who knows out of experience, from the inside, how volatile, how disruptive, how violent, how misogynistic is the culture of Islam today and has been during my lifetime, and how it greatly threatens our liberal democracy that I cherish, since I know what is its opposite."
But for some "lighter fare" that speaks to your "the Woke are in love with Islam" 🙂, you might be "amused" by this old YouTube video on "Feminists Love Islamists":
Why did science fizzle out in some places, and grow in Europe? I think it is explained by the book Why Nations Fail. Inclusive/pluralistic/free-market economies thrive on innovation; acquisitive-run-by-elite economies do not like innovation and generally suppress it. I'm re-reading the book after learning more about history, and appreciate it more than ever
Personally I'm leery of economists ventruing into fields they rarely understand, such as cultural anthropology, psychology, sociology and history, but as a complement to such studies 'Why Nations Fail' adds a necessary perspective, despite it being biased towards the current neoliberal capitalist model.
If you haven't already, do read Fukuyama's 'The Origins of Political Order' and 'Political Order and Political Decay' where he delves into the question "Why aren't all nations like Denmark?"
(Of course, him being no idiot and full well knowing the answer, he lets the reader figure out the quiet part for themselves, without being didactic about it.)
“Scholarship in the Muslim world around the ti?me of al-Haytham was much more advanced than anything that existed in Europe at the time, but it kind of fizzled out and never really went anywhere. I don’t know why.” Ian McGilchrist has some suggestions - after a time human brains get more interested in process, algorithms, things, than in ideas and creativity ie repeating what the last guy / gal did, until we’ve exhausted the initial creative idea (eg the solution to the square root of -1) and we now need a fresh idea. Don’t we just!?
I don't know that this idea expresses it right. Certainly the people I used to work with were all interested in coming up with new ideas and new ways of looking at things.
But there were pressures. The pressure to publish, for example, meant that there was a strong temptation, or push, to just "crank the handle" a bit, simply to get your publication rate looking 'good enough' to survive your next performance review.
I wish you'd been my calculus instructor. Even though it's been a while, it is making so much better sense with your explanations (in this case differentiation).
I both love and hate trying to get across these ideas. The 'hate' is because I want *everyone* to really understand them and feel a sense of failure when I can't manage to help someone understand things.
Now that we have learned how the exp function managed to escape from the oppressive realarchy, I only have to find out what a Trilby is.
P.S.: Sometimes substack displays embedded LaTeX with a "shadow", i.e., with the underlying command (like "\(\exp(iy) = \cos(y) + i \sin(y)\)") right behind the formula. Maybe the shadow will vanish if I reload the page.
Radar engineers use what is known in the discipline as Inphase/Quadrature notation to represent signal phase and amplitude data.
And whaddya know...phase and amplitude can be represented as complex numbers, and everything just works, as if by magic. It's amazing to see "pure" mathematical concepts invented hundreds of years ago finding such tremendous relevance and applicability in all sorts of technical domains today.
Trivia I remember: There is only one number that, when raised to the power x, satisfies the differential equation dy/dx = y. That number was (per the lecture I barely remember) was first described not by a mathematician, scientist or engineer, but a banker. The banker noticed that as the bank shortened the interval for compounding interest, it got more money--but to a limit that he named "e". Or something like that.
Well, that gave me terrible flashbacks to 1st year Pure Maths at uni (way back in the Cretaceous Period). Invent i to be the square root of minus one. As for lemmas where you assume what you want to prove and then prove it by assuming the proof (I think that is how those things went but it was so long ago. I fled Maths after that and stuck to English Lit and Psychology and Philosophy. A weird species, mathematicians!
😂
You're probably talking about a proof method called "proof by contradiction" (there's a fancy latin name: modus bonehead, or something like that). You *assume* something is true, then work out the consequences of that assumption being true. If these new consequences you derive are inconsistent (eg they lead to something like 1 = 2) then it means your original assumption is false.
An example of the kind of assumption made might be "assume any odd number can be factored by 2" - you could then show that this assumption leads to a nonsensical result and so the original assumption must be wrong: you've then shown that an odd number can't be (perfectly) divided by 2. This is a bit of a silly example to get the idea across.
Choosing the qualifier is important - you might want to phrase it better as "assume there exists one odd number that can be perfectly divided by 2" and then show that this assumption leads to a contradiction, thereby proving that there exists no such odd number.
Reductio ad absurdem
Yes, that sounds more like it. Trouble is, as I get older, each new bit of knowledge that gets into my brain pushes out an older bit of knowledge, so my memory of Uni Maths all those decades ago gets seriously degraded.
Thanks for the explanation.
Mathematician joke...
Did you know that 1 + 1 = 3, for very large values of 1, and very small values of 3?
Thanks...I'll be here all week...please tip the waitress.
The reason the budding scholarship in the islamic world petered out was because it came into conflict with the priesthood, to put it bluntly.
To the priests, even a thousand years ago, the Quran contained all necessary knowledge since it was the word of god by way of the angel who dictated to the prophet who, being illiterate, repeated it to a scribe.
(The kadis and imams of the time had obviously never heard of 'chinese whispers'.)
This set-in-stone approach was very popular with the secular leaders of the islamic world, and mainly being arabs or influenced by arabs and therefore lacking any scientific or even civilisational basis (the arabs' achievements were mainly repossessed greco-roman-egyptian ones in new packaging, and being smart enough to use clever conquered peoples wok to the benefit of the state) above tribal level, they opted for the priests' line of thinking, leading to islamic scholars either renouncing their idas, getting murdered or moving to China or Europe.
(There's a lesson in there for parlour-pinks, wokesters and sundry.)
The above is also the reason why Europe started surpassing the islamic world after the Black Plague had done its job: here, new ideas were welcome as long as they didn't challenge papal dogma, and the catholic church virtually killed its own moral authority by its - even at the time! - well-known depravity, until Luther popped the zit called Pope. Had the islamic world produced a similar figure as Martin Luther... but islamic-arabic culture is incapable of that it seems, being firmly rooted in clan-structures and inbreeding (yes, marrying first cousins for ten centuries is inbreeding, it's not racism but genetics and genetics doesn't care about feelings).
Again, there's a lesson: don't muck about with anything that has to do with fertility, hormones and heritability.
Speaking of exponential growth: a wild sow can have 2-3 litters per year, of 6-8 "boarlets". Without predators, almost all of them will reach maturity inside a year and start reproducing.
Q: Starting with 100 females and 100 males (Yes, Virginia, there are only two sexes) and assuming no more than 10% of "boarlets" die before reproducing, and that all survivors reach at least 5 years of age; when do the population of boar reach 100 000 adult animals?
The above is too difficult for 100% of all feminists, LGBTP, Greens, woke, liberals, progressives and modern-day po-mo marxists I've asked.
Which is the reason Sweden have a very real problem with far more than 100 000 boars destroying crops and gardens.
Science doesn't care about fee-hee-hee-lings.
Yes - that's my speculation too. I suspect that the religious 'scholars' were more influential than the 'academic' scholars.
We've sort of witnessed something a bit similar with regards to covid. You can see it "between the lines", so to speak, where some authors published some result that was clearly against the 'narrative' when you examined it carefully, but the abstract and conclusions were written so as to appear to *support* the narrative.
We have it in the GenderWoo field where it's next to impossible to get 'honest' research funded.
Even the prestigious journal Nature ran an editorial basically stating that some research should be rejected on *moral* grounds (i.e. if it supported a conclusion or looked to be investigating something that might upset the Woke wagon).
Uh-oh. (Pause to get another cup of coffee and reread.)
Ok, now that I’m caffeinated, I’ve decided I am enjoying your “complex” series. Human intellect need not be cowed by the boundaries of what first appears rational or possible. (Even if I backed slowly away from some of the math you presented like it was a transwoman in a thong.) Instead of reflexively mocking crazy-seeming ideas, we can test them— and that gives us the square root of negative 1. (But not “oh, look, that transwoman in a thong is pregnant!”) We want to do this; there is something surprising and indeed delightful about being able to. But when there is no sense to be found, only nonsense, well… back to mocking, one hopes. (Unless one is the transwoman with a beer gut. Who is brave and beautiful and I hope doesn’t come after me because she is a foot taller than me and a hundred pounds heavier.)
Euler said it best. e^i*pi = -1. an incredible unification of the complex, exponential, unitary, geometric, negative and if rewritten, zero. Greatest equation of all time?
Yes - that equation is really quite something isn't it?
Pure magic
Rudolph: Scholarship in the Muslim world around the time of al-Haytham was much more advanced than anything that existed in Europe at the time, but it kind of fizzled out and never really went anywhere. I don’t know why.
Good question. For a bit of a long winded answer, though still a fascinating one, see this oldish essay in "The New Atlantis" on "Why the Arabic World Turned Away from Science". As another commenter suggested or argued, part of the reason was that science conflicted with -- STILL conflicts with --religious dogma. Though there are still some other interesting aspects:
"There is a final reason why it makes little sense to exhort Muslims to their own past: while there are many things that the Islamic world lacks, pride in heritage is not one of them. What is needed in Islam is less self-pride and more self-criticism. Today, self-criticism in Islam is valued only insofar as it is made as an appeal to be more pious and less spiritually corrupt. And yet most criticism in the Muslim world is directed outward, at the West. This prejudice — what Fouad Ajami has called (referring to the Arab world) 'a political tradition of belligerent self-pity' — is undoubtedly one of Islam’s biggest obstacles. It makes information that contradicts orthodox belief irrelevant, and it closes off debate about the nature and history of Islam."
https://www.thenewatlantis.com/publications/why-the-arabic-world-turned-away-from-science
"belligerent self-pity" -- a rather damning indictment. Not just of Islam, but, maybe arguably, much of the Woke as well.
Thanks for that Steersman - I'll take a peek.
I really like that expression "belligerent self-pity" which does seem to capture a fair bit of the 'woke' attitude.
Thank you for digging that one up, it's a really good long-form article with great references!
"But the Islamic turn away from scholarship actually preceded the civilization’s geopolitical decline — it can be traced back to the rise of the anti-philosophical Ash’arism school among Sunni Muslims, who comprise the vast majority of the Muslim world."
This quote really sums it up, and much better than my rambling atempt earlier.
"While the Mu’tazilites had contended that the Koran was created and so God’s purpose for man must be interpreted through reason, the Ash’arites believed the Koran to be coeval with God — and therefore unchallengeable. At the heart of Ash’ari metaphysics is the idea of occasionalism, a doctrine that denies natural causality. Put simply, it suggests natural necessity cannot exist because God’s will is completely free. Ash’arites believed that God is the only cause, so that the world is a series of discrete physical events each willed by God."
This is how virtually all moslems reason today, which makes it completely impossible to "integrate" them or for them to integrate with any other society or civilisation; whatever scientific qeury you have, in any filed of scence, the answer is always "God wills it" anyway.
The number of times I've heard and seen arab children in Sweden, iraqis and palestinians especially, answer with a shrug and a "Because God" to any question relating to science is in three digits.
"Why does the sky change colour at dawn and at evening?"
"Bcause God wants it to."
Completely serious, no sarcasm.
Small wonder the Woke are in love with islam (or rather their fantasy-version of it.)
Rikard: "Thank you for digging that one up ...."
De nada; share the wealth; praise the lord & pass the ammunition ... 🙂. I've gotten a fair amount of "mileage" out of it so I like to pass it around whenever I get the chance.
Rikard: "... completely impossible to 'integrate' them or for them to integrate with any other society or civilization ..."
Amen to that. Muslims generally don't play well with anyone -- as I think many countries, Sweden in particular, are finding to their "chagrin" or misfortune. But somewhat apropos of which, see my 2018 archived article in Canada's Post Millennial asking, "Can Islam, Sharia and a Secular Democracy coincide?" -- though I had titled my submission to them using "coexist" instead of "coincide":
https://web.archive.org/web/20180405214735/https://www.thepostmillennial.com/islam-sharia-secular-democracy/
Of particular note therein is a quote of Muslim reformer Shireen Qudosi who said, in an article of hers at The Blaze, "either Islam needs to evolve or it needs to die" -- don't see much evidence of the former.
But for another bit of Canadiana on the Muslim front -- which I apparently wasn't aware of at the time I wrote that article, see this quote of "Salim Mansur, professor of political science at the University of Western Ontario" in his rather damning testimony to Canada's "Standing Committee On Citizenship And Immigration" in 2012:
Mansur: "We have the precedent of how we selectively closed immigration from the Soviet bloc countries during the Cold War years, and we need to consider doing the same in terms of immigration from Muslim countries for a period of time given how disruptive is the cultural baggage of illiberal values that is brought in as a result. ....
Lest any member wants to instruct me that my views are in any way politically incorrect, or worse, I would like members to note that I come before you as a practising Muslim who knows out of experience, from the inside, how volatile, how disruptive, how violent, how misogynistic is the culture of Islam today and has been during my lifetime, and how it greatly threatens our liberal democracy that I cherish, since I know what is its opposite."
https://www.ourcommons.ca/DocumentViewer/en/41-1/CIMM/meeting-51/evidence#Int-7696927
But for some "lighter fare" that speaks to your "the Woke are in love with Islam" 🙂, you might be "amused" by this old YouTube video on "Feminists Love Islamists":
https://www.youtube.com/watch?v=ecJUqhm2g08
Certainly some parallels in the worst of both.
Why did science fizzle out in some places, and grow in Europe? I think it is explained by the book Why Nations Fail. Inclusive/pluralistic/free-market economies thrive on innovation; acquisitive-run-by-elite economies do not like innovation and generally suppress it. I'm re-reading the book after learning more about history, and appreciate it more than ever
Personally I'm leery of economists ventruing into fields they rarely understand, such as cultural anthropology, psychology, sociology and history, but as a complement to such studies 'Why Nations Fail' adds a necessary perspective, despite it being biased towards the current neoliberal capitalist model.
If you haven't already, do read Fukuyama's 'The Origins of Political Order' and 'Political Order and Political Decay' where he delves into the question "Why aren't all nations like Denmark?"
(Of course, him being no idiot and full well knowing the answer, he lets the reader figure out the quiet part for themselves, without being didactic about it.)
“Scholarship in the Muslim world around the ti?me of al-Haytham was much more advanced than anything that existed in Europe at the time, but it kind of fizzled out and never really went anywhere. I don’t know why.” Ian McGilchrist has some suggestions - after a time human brains get more interested in process, algorithms, things, than in ideas and creativity ie repeating what the last guy / gal did, until we’ve exhausted the initial creative idea (eg the solution to the square root of -1) and we now need a fresh idea. Don’t we just!?
I don't know that this idea expresses it right. Certainly the people I used to work with were all interested in coming up with new ideas and new ways of looking at things.
But there were pressures. The pressure to publish, for example, meant that there was a strong temptation, or push, to just "crank the handle" a bit, simply to get your publication rate looking 'good enough' to survive your next performance review.
I wish you'd been my calculus instructor. Even though it's been a while, it is making so much better sense with your explanations (in this case differentiation).
Thanks John
I both love and hate trying to get across these ideas. The 'hate' is because I want *everyone* to really understand them and feel a sense of failure when I can't manage to help someone understand things.
Now that we have learned how the exp function managed to escape from the oppressive realarchy, I only have to find out what a Trilby is.
P.S.: Sometimes substack displays embedded LaTeX with a "shadow", i.e., with the underlying command (like "\(\exp(iy) = \cos(y) + i \sin(y)\)") right behind the formula. Maybe the shadow will vanish if I reload the page.
Radar engineers use what is known in the discipline as Inphase/Quadrature notation to represent signal phase and amplitude data.
And whaddya know...phase and amplitude can be represented as complex numbers, and everything just works, as if by magic. It's amazing to see "pure" mathematical concepts invented hundreds of years ago finding such tremendous relevance and applicability in all sorts of technical domains today.
If anybody's interested...
https://en.m.wikipedia.org/wiki/In-phase_and_quadrature_components
Or in more layman's terms...
https://www.skyradar.com/blog/an-introduction-to-i/q-signals
Oh yes, the quadratures and heterodyne and homodyne detection 😂
I remember them well from my early days looking at so-called "squeezed" states of light.
Trivia I remember: There is only one number that, when raised to the power x, satisfies the differential equation dy/dx = y. That number was (per the lecture I barely remember) was first described not by a mathematician, scientist or engineer, but a banker. The banker noticed that as the bank shortened the interval for compounding interest, it got more money--but to a limit that he named "e". Or something like that.