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# Making the impossible possible

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This isn't how it was. There were methods that gave "exact" results before calculus.

posted by Obscure Reference at 5:40 AM on May 25, 2012

No, it really is impossible. The square root of X is the number that, when multiplied by itself, gives X. Since a negative times a negative is always a positive, there is no number that will work.

It is the same thing as the invention of negative numbers. You can't have a negative number of things, but you need negative numbers to be able to make calculations. Just as 4 - 3 is the same as 4 + -3, so is 4 + sqrt(-1) the same as 4 +

posted by gjc at 7:26 AM on May 25, 2012

Sure there is; in fact, there are typically two

Asking whether it "really is impossible" to extract the square root of -5 misses the point. As G.H. Hardy said,

"…it does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition. It was not a triviality even to the greatest mathematicians of the eighteenth century. They had not the habit of definition: it was not natural to them to say, in so many words, `by X we mean Y.’ … it is broadly true to say that mathematicians before Cauchy asked not ‘How shall we define

1 – 1 + 1 – 1 + …

but

‘What is 1 – 1 + 1 – 1 + …

and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal."

I emphasize that this really is part of my answer to the question, because Hardy is pointing here exactly to the kind of conceptual move that makes possible phenomena like the one OP described.

posted by escabeche at 7:39 AM on May 25, 2012 [2 favorites]

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# Making the impossible possible

May 24, 2012 6:13 PM Subscribe

Help with a simultaneously math & non-math question: Finding the square root of a negative number was impossible until imaginary numbers came along. What are examples of other things that were impossible until we altered systems that have been created?

I know this is pretty opened ended but it's for an essay about things that were previously thought to be impossible. I'm looking for:

1) things that were once thought impossible but then someone invented something that made it possible (got some but could use more).

2) things that were considered impossible but someone changed a system to make it so. Most of my examples are pretty serious (ie: Rosa Parks) and I can use some more silly & funny. Any help or ideas would be much appreciated, even if you don't think it's so hot - it may help with the direction of story. Thanks.

I know this is pretty opened ended but it's for an essay about things that were previously thought to be impossible. I'm looking for:

1) things that were once thought impossible but then someone invented something that made it possible (got some but could use more).

2) things that were considered impossible but someone changed a system to make it so. Most of my examples are pretty serious (ie: Rosa Parks) and I can use some more silly & funny. Any help or ideas would be much appreciated, even if you don't think it's so hot - it may help with the direction of story. Thanks.

Running a four-minute mile.

Splitting the atom.

posted by spikeleemajortomdickandharryconnickjrmints at 6:38 PM on May 24, 2012

Splitting the atom.

posted by spikeleemajortomdickandharryconnickjrmints at 6:38 PM on May 24, 2012

Zero? I can imagine people beforehand asking something like, "How can you count something that doesn't exist?" (oops, croutonsupafreak beat me.)

As a vaguely similar concept, before the discovery of industrial chemical processes necessary to produce it en masse in the late 19th century, aluminum was more valuable than gold. So, I would think that someone of the earlier era would find the idea of wrapping a piece of meat in aluminum foil and throwing it in a fire to cook somewhat startling.

Cheese, especially things like blue cheese, would probably seem bizarre to anyone from a culture that didn't have the technology down: simply the idea that you could eat something like that without it killing you.

posted by XMLicious at 6:45 PM on May 24, 2012 [1 favorite]

As a vaguely similar concept, before the discovery of industrial chemical processes necessary to produce it en masse in the late 19th century, aluminum was more valuable than gold. So, I would think that someone of the earlier era would find the idea of wrapping a piece of meat in aluminum foil and throwing it in a fire to cook somewhat startling.

Cheese, especially things like blue cheese, would probably seem bizarre to anyone from a culture that didn't have the technology down: simply the idea that you could eat something like that without it killing you.

posted by XMLicious at 6:45 PM on May 24, 2012 [1 favorite]

Mathematics is full of episodes like this. Measuring many geometric quantities, like the hypotenuse of a right triangle or the circumference of a circle, was in some sense impossible until people enlarged their concept of "number" to include the quantities we would now call "irrational." (I say "in some sense" since people were certainly capable of laying a measuring stick along the hypotenuse of a real-world triangle and making their best estimate.) People weren't able to speak quantitatively about uncertain events until the theory of probability was developed. People weren't able to describe the difference between two infinite quantities until Cantor developed the theory of transfinite cardinals in the 19th century. People weren't able to speak quantitatively about the extent of association between two phenomena until Galton and Pearson developed the notion of correlation around the same time. And so on, and so on.

posted by escabeche at 7:09 PM on May 24, 2012 [2 favorites]

posted by escabeche at 7:09 PM on May 24, 2012 [2 favorites]

Hre's a bunch of statements of the impossibility of things that might help

posted by Sparx at 7:23 PM on May 24, 2012

posted by Sparx at 7:23 PM on May 24, 2012

Godel's incompleteness theorem was quite effective that in logical systems there will always be things that can be proven neither true nor false.

posted by plinth at 7:27 PM on May 24, 2012

posted by plinth at 7:27 PM on May 24, 2012

The old FedEx commercial Evolution of Delivery shows shifting expectations of communications and logistics, ending with "I need it tomorrow." At Internet speed, even this seems quaint.

posted by Cool Papa Bell at 7:28 PM on May 24, 2012

posted by Cool Papa Bell at 7:28 PM on May 24, 2012

I think you misunderstand the question you yourself are asking.

It wasn't "impossible" to take the square root of a negative number, it was simply that it was against the rules. Within the definition of that particular system, "square root" didn't apply to negative numbers.

posted by Chocolate Pickle at 8:26 PM on May 24, 2012

It wasn't "impossible" to take the square root of a negative number, it was simply that it was against the rules. Within the definition of that particular system, "square root" didn't apply to negative numbers.

posted by Chocolate Pickle at 8:26 PM on May 24, 2012

Funny that you should speak of imaginary numbers.

The so-called real numbers (roughly, the set of all non-terminating decimal fractions) have the property of being

When you extend the real numbers to the set of complex numbers, you lose the property of total order. There is no natural "order" relation over the complex numbers.

If you extend the set of complex numbers to the set of quaternions, it suddenly becomes clear that not only are these numbers without an order, but they're also

Finally, if you extend the set of quaternions to the set of octonions, you lose the property of

At this point you have a set of numbers that exhibit none of the properties that you would expect numbers to have.

posted by Nomyte at 9:27 PM on May 24, 2012 [5 favorites]

The so-called real numbers (roughly, the set of all non-terminating decimal fractions) have the property of being

*totally ordered*. Given any two real numbers, we can easily compare their magnitude.When you extend the real numbers to the set of complex numbers, you lose the property of total order. There is no natural "order" relation over the complex numbers.

If you extend the set of complex numbers to the set of quaternions, it suddenly becomes clear that not only are these numbers without an order, but they're also

*non-commutative*. A times B is not necessarily the same as B times A.Finally, if you extend the set of quaternions to the set of octonions, you lose the property of

*associativity*. In other words, (A times B) times C is not necessarily equal to A times (B times C).At this point you have a set of numbers that exhibit none of the properties that you would expect numbers to have.

posted by Nomyte at 9:27 PM on May 24, 2012 [5 favorites]

It used to be impossible for same-sex couples to marry.

posted by flabdablet at 10:26 PM on May 24, 2012

posted by flabdablet at 10:26 PM on May 24, 2012

Speaking of chemistry, it was once thought that organic chemicals came from living processes (hence the name) and inorganic came from nonliving processes, and that there could be no crossover, until Friedrich Wöhler synthesized a few.

For a while it was thought that there could never be more than three American television networks (like there can never be more than two political parties). Then Fox showed up. (Then broadcast TV declined to semi-irrelevance, but that's a different matter.)

posted by hattifattener at 10:27 PM on May 24, 2012 [1 favorite]

For a while it was thought that there could never be more than three American television networks (like there can never be more than two political parties). Then Fox showed up. (Then broadcast TV declined to semi-irrelevance, but that's a different matter.)

posted by hattifattener at 10:27 PM on May 24, 2012 [1 favorite]

Non-Euclidean geometries come to mind.

posted by kickingtheground at 11:08 PM on May 24, 2012

posted by kickingtheground at 11:08 PM on May 24, 2012

>

No, I think this is exactly what they're asking. What are "impossible" things that were not impossible at all, they just fell outside of the rules-as-currently-understood.

posted by desuetude at 11:39 PM on May 24, 2012 [1 favorite]

*It wasn't "impossible" to take the square root of a negative number, it was simply that it was against the rules. Within the definition of that particular system, "square root" didn't apply to negative numbers.*No, I think this is exactly what they're asking. What are "impossible" things that were not impossible at all, they just fell outside of the rules-as-currently-understood.

posted by desuetude at 11:39 PM on May 24, 2012 [1 favorite]

It is still impossible (ie: undefined) to take the square root of a negative number when dealing with some "kinds of numbers" (ie: topologies). When dealing with the space of integers, or rationals, or the incredibly misnamed Real Numbers square roots of negatives are as undefined today as they ever were.

People did not change the rules for these existing spaces. They just invented a new space with different properties where square roots of negatives are defined.

If you consider the space known as a Riemann Sphere then division by zero is defined. Doesn't mean you can divide by zero with Real Numbers or Complex Numbers (the actual name for "imaginary" numbers)

posted by Riemann at 11:56 PM on May 24, 2012 [2 favorites]

People did not change the rules for these existing spaces. They just invented a new space with different properties where square roots of negatives are defined.

If you consider the space known as a Riemann Sphere then division by zero is defined. Doesn't mean you can divide by zero with Real Numbers or Complex Numbers (the actual name for "imaginary" numbers)

posted by Riemann at 11:56 PM on May 24, 2012 [2 favorites]

I was talking to a friend once in a bar about Marvell's poem The Definition of Love because I liked the idea of perfect lovers being like parallel lines that 'though infinite can never meet.'

She was a doing some grad Maths or something and proceeded to blow my tiny boozed mind with some idea from projective geometry where apparently parallel lines can theoretically meet.

I'm fuzzy on the actual details, but I thought that was pretty cool.

posted by man down under at 1:40 AM on May 25, 2012

She was a doing some grad Maths or something and proceeded to blow my tiny boozed mind with some idea from projective geometry where apparently parallel lines can theoretically meet.

I'm fuzzy on the actual details, but I thought that was pretty cool.

posted by man down under at 1:40 AM on May 25, 2012

Calculus. Until this was perfected in the seventeenth century, it was impossible to calculate the area under a curve. You could approximate it with the method of exhaustion, i.e., divide it up into a bunch of rectangles and ignore the bits that didn't quite fit, but (1) this was an enormous pain in the ass, and (2) didn't actually give you the area, just a more-or-less accurate approximation thereof.

Similarly, non-Euclidean geometries are... weird, but important. Euclid's fifth postulate, i.e., that parallel lines never converge, is something that mathematicians tried to prove for two thousand years. Some guys went pretty close to nuts trying to work it out. Eventually, people who started trying to prove it by assuming that parallel lines

posted by valkyryn at 4:02 AM on May 25, 2012 [1 favorite]

Similarly, non-Euclidean geometries are... weird, but important. Euclid's fifth postulate, i.e., that parallel lines never converge, is something that mathematicians tried to prove for two thousand years. Some guys went pretty close to nuts trying to work it out. Eventually, people who started trying to prove it by assuming that parallel lines

*did*cross... and found out that the resulting systems, hyperbolic and elliptical geometry, were consistent. This gives*really*counter-intuitive results, because our brains are pretty hard-wired to think in Euclidean terms. But, for example, while the sum of the interior angles of a triangle in Euclidean space is always 180 degrees, in elliptical geometry it's always more than that, though exactly how much varies depending on what kind of triangle you're drawing. Could be anything from just a hair over 180 to just a hair under 360. On the other hand, in hyperbolic space, triangles always have interior angles that measure*less*than 180.posted by valkyryn at 4:02 AM on May 25, 2012 [1 favorite]

*Calculus. Until this was perfected in the seventeenth century, it was impossible to calculate the area under a curve. You could approximate it with the method of exhaustion, i.e., divide it up into a bunch of rectangles and ignore the bits that didn't quite fit, but (1) this was an enormous pain in the ass, and (2) didn't actually give you the area, just a more-or-less accurate approximation thereof.*

This isn't how it was. There were methods that gave "exact" results before calculus.

posted by Obscure Reference at 5:40 AM on May 25, 2012

*It wasn't "impossible" to take the square root of a negative number, it was simply that it was against the rules. Within the definition of that particular system, "square root" didn't apply to negative numbers.*

No, it really is impossible. The square root of X is the number that, when multiplied by itself, gives X. Since a negative times a negative is always a positive, there is no number that will work.

*i*was just invented as a placeholder/handwave so that one could complete the intermediate calculations to get to the desired result.

It is the same thing as the invention of negative numbers. You can't have a negative number of things, but you need negative numbers to be able to make calculations. Just as 4 - 3 is the same as 4 + -3, so is 4 + sqrt(-1) the same as 4 +

*i*

posted by gjc at 7:26 AM on May 25, 2012

*Since a negative times a negative is always a positive, there is no number that will work.*

Sure there is; in fact, there are typically two

*different*numbers that are square roots of X. If you want to insist that i, and for that matter, -3, are not numbers because you can't have an imaginary or negative "number of things," that's your right, but then your notion of number is going to be pretty impoverished, missing not only -3 and i but also the square root of 2 and 1/2.

Asking whether it "really is impossible" to extract the square root of -5 misses the point. As G.H. Hardy said,

"…it does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition. It was not a triviality even to the greatest mathematicians of the eighteenth century. They had not the habit of definition: it was not natural to them to say, in so many words, `by X we mean Y.’ … it is broadly true to say that mathematicians before Cauchy asked not ‘How shall we define

1 – 1 + 1 – 1 + …

but

‘What is 1 – 1 + 1 – 1 + …

and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal."

I emphasize that this really is part of my answer to the question, because Hardy is pointing here exactly to the kind of conceptual move that makes possible phenomena like the one OP described.

posted by escabeche at 7:39 AM on May 25, 2012 [2 favorites]

It sounds to me like what you're asking for is examples of Paradigm Shifts.

This is the notion (as put forward by the epistemologist Thomas Kuhn) that certain discoveries completely change the scientific landscape and you can never go back!

The big example of this is relativity.

There are a lot of good examples in that Wikipedia link, but in a lot of cases the thing they make possible is thinking about the universe in a more accurate way.

Sadly the term Paradigm shift has been grievously misused by marketers and businessy folk, which may make googling a little tricky in some places. Imaginary Numbers, Calculus and Zero are all major paradigm shifts in maths though.

posted by Just this guy, y'know at 7:44 AM on May 25, 2012

This is the notion (as put forward by the epistemologist Thomas Kuhn) that certain discoveries completely change the scientific landscape and you can never go back!

The big example of this is relativity.

There are a lot of good examples in that Wikipedia link, but in a lot of cases the thing they make possible is thinking about the universe in a more accurate way.

Sadly the term Paradigm shift has been grievously misused by marketers and businessy folk, which may make googling a little tricky in some places. Imaginary Numbers, Calculus and Zero are all major paradigm shifts in maths though.

posted by Just this guy, y'know at 7:44 AM on May 25, 2012

Thanks everyone. These all helped a lot.

posted by CaptainCaseous at 10:02 AM on June 28, 2012

posted by CaptainCaseous at 10:02 AM on June 28, 2012

This thread is closed to new comments.

Flight in general.

Achieving the speed of sound.

Knowing the sex of your child before it is born.

posted by HuronBob at 6:33 PM on May 24, 2012