Hey, Vsauce. Michael here. The title of this video is misspelled in honour of mistakes. Mistakes are everywhere, they surround us like air To err is human. Faults, flaws, faux pas, fumbles and fallacies are as much a part of who we are today as the stuff we've gotten right. For instance, if a knight knocked your knuckle or knifed your knee, why would there be so many "k"s? Well, the "k"s are silent by mistake. The original old English forms of these
words were pronounced with "k" sounds. K-nife, k-night, k-nee. But that's a bit of a mouthful to say and
because reading and writing weren't as common hundreds of years ago, people just pronounced words the way
they want to do, regardless of spelling. Spelling is fun. Warren G knows what
I'm talking about. In the fourth verse of his "What's Next", he asks what's next. What's next, what's N-X-E-T Spelling isn't the only thing we get wrong. The history of science is a graveyard of dead an abandoned ideas.
Fritz Machlup coined the phrase "Half-life of knowledge". The amount of time
it takes for half of the knowledge within a field to be superseded by new, better ideas
or to simply be shown untrue. Donald Hebb famously estimated that the half-life of knowledge in psychology is just five years. Humans are awesome, don't get me wrong,
but we tend to believe that what we currently think we know about the universe is reasonably correct, even though statistics aren't on our side. Previous generations incorrectly thought the exact same thing
about what they used to think was true. My favourite examples of the ubiquity
of mistakes are production errors in popular songs. They're like humbling Easter eggs, just waiting
to be found. For instance, Buffalo Springfield's "For What It's Worth". 26 second in the "be" in the lyric "beware" is clipped, as is the "p" in "people" at 1:24 minute. Once you hear the mistake, it's tough to unhear it. In "Hey Jude" Paul McCartney misses a cord. And if you
listen closely you can clearly hear him say "F***ing h*ll". Seriously, it's in the actual song. Go listen to it. In fact, there's an entire
website that chronicles mistakes in Beatles' songs. Take a look at this 700-year-old prayer book. A monk wrote it on fresh, clean paper. I'm kidding, of course. He scraped the ink
off an older manuscript, cut the pages and rotated them before writing all over them. A mistake? Well, kind of. Recent multispectral images of the prayer book have revealed that
the old manuscript he erased was a previously unknown copy of a work by Archimedes. It was called "The Method" and laid out the heart of calculus thousands of years before Newton and Leibniz. If that one monk hadn't erased that one book, would we be hundreds, thousands of years
mathematically and technologically more advanced today than we currently are? It's hard to say. All that is certain is that we would
continue to make ridiculous mistakes, like the Mars Climate Orbiter.
This 327.6 million dollar expedition burned up in the red planet's atmosphere because when calculating flight manoeuvres NASA
used the agreed-upon metric units while Lockheed Martin used the imperial system. This is Neil Armstrong taking humankind's first steps on the Moon. It's about the best footage we have. The original tapes containing
the highest quality recording of that moment have been lost. They were probably recorded over by later test missions. Ten years ago Sergio Martinez became lost in the woods while hunting outside of San Diego.
Hoping to attract the attention of rescuers, he lit a small fire. But that fire quickly got out of control
and became a giant Cedar Fire. It destroyed 300,000 acres of land, 2,322 homes and killed 15 people. The man carrying a wounded soldier in
this painting, based on a photograph taken during World War I, is Henry Tandey, an English recipient
of the Victoria Cross. Four years after the event in this image
Tandey caught a wounded German soldier in his gun sights.
But rather than kill the man, Tandey took pity on his wounded state
and spared his life. The German he allowed to live was this man. Later the man whose life he
spared wore his mustache shorter but still had the same name, Adolf Hitler. In 1918 did Henry Tandey miss a chance to kill
Hitler? Detailed researchers found that the exact days their units were in the same
location don't quite match up. The story is apocryphal but what is known is that Hitler owned a copy of the painting of Tandey, and in 1938, when meeting with Neville Chamberlain,
Hitler pointed to Tandey and told Chamberlain: "That man came
so close to killing me in 1918 that I thought I
should never see Germany again." So, who's wrong? Maybe Hitler confirmed the story merely because he hoped to make up extra
evidence that providence had kept him alive to pursue his goals. Either way, someone is mistaken. Missed opportunities are a bummer.
Obsessing over them is not healthy but regret is a powerful emotion. How do you deal with regret, with guilt? Can you? In the early morning hours on a
bench outside a hotel in Anaheim this summer Ze Frank told me something
I am going to paraphrase. I love this metaphor. Stuff in your past is like a carving on the bark of sapling. Over time, the scar, the carving won't go away. Because of the way trees grow it won't
go up or down much either, it'll just stay right where it began. It might even get darker. But it won't get bigger. You, however, can. You can keep growing,
doing more things, more branches, being more things. The wound won't get
smaller but you can make it a smaller part of who you are. Maybe regrets are like that. They stick around forever like arborglyphs. Or maybe they make like a tree and leaf. A red or purple leaf in the autumn.
As days get shorter and chlorophyll production decreases, the yellow and orange carotenoids, which are always in leaves, appear as the green fades.
But red and purple leaves are the interesting ones.
As winter approaches it would seem to be a good time for trees to conserve energy but some trees do the opposite. Instead of giving up, they spend extra energy producing anthocyanins to turn their leaves red and purple hues. These colours protect their leaves from
sun damage before their nutrients can all be used and may also be a defence
against insects looking for a parasitic home.
A way for the tree to tell the insects: "Yes, I am in part dying but not without a fight. I am still very much vital." Drought and even turn a kit applied by man can bring about these colors prematurely.
When you look at beautiful autumn colours, you are looking at stress. But the bigger the fight the trees
put up, the more energy they put into their defences at the
very end, the more brilliant their colours will be. Winter will eventually come. But scientifically, the brightest, deepest, most remarkable colours come from not giving up too easily or quickly. And as always, thanks for watching.
It's clear from some of his geometric work (the "method of exhaustion" in particular) that Archimedes had some notion that, if you can approximate something up to arbitrarily small error, you essentially know how to calculate that thing. But a lot of essential ideas of calculus just didn't exist at the time. Forget differentiation as an inverse process, Archimedes wasn't even working with the basic ideas of functions, equations, and graphs that you'd learn in a basic algebra course today.
I would say that Archimedes (and others besides) probably understood the basic idea of calculus, but the language just wasn't there to describe it properly as Newton and Leibnitz did, and certainly not to properly formalize it as was later done.
If there had been a stronger foundation for calculus, there's a very good chance that Newton would have been centuries too late to invent it. Even if it was just a spark, the progression of human understanding of any mathematical field is exponential from its inception. If the calculus were already formed by the time Newton was around, the shoulders of the giants he stood on would be a lot taller, and who knows what we might have known by now. But I think this isn't really worth considering, other than to praise the already revered Archimedes.
I think many ancient mathematicians danced around the idea of limits (e.g. Archimedes's approximations of pi, Zeno's paradoxes), but lacked the formal language to deal with it. I would hesitate to say that this means that any of them were on the verge of calculus because I'm not sure I'd classify "calculus" as simply understanding limits. I think calculus is really the study of functions by way of limits, and functions just weren't really a concept until several hundred years later.
It's accurate to say he used calculus concepts, but he was nowhere near our full understanding of differentiation, integration, and the relationship between them. So it doesn't make a lot of sense to say he "understood calculus."