During the Enlightenment, the world was viewed as an orderly place in which everything operated by precise mathematical principles. Beginning with the Romantic backlash, though, and unfolding into the early 20th century, rationalism in mathematics, as well as the concept of absolute truth, was called into question.

This change came about through startling discoveries in the field of mathematics, revealing that what we thought were fundamental principles in both geometry and arithmetic were not always true in every situation.

This is a transcript from the video seriesRedefining Reality: The Intellectual Implications of Modern Science. Watch it now, on The Great Courses Plus.

Interestingly, this collapse of certainty—and the resulting wide-scale wreckage to the foundations of our reason—was reflected in a pair of notable and related works of fiction coming out of Britain in the late 1800s: Lewis Carroll’s *Alice’s Adventures in Wonderland*, and Edwin Abbott Abbott’s *Flatland: A Romance of Many Dimensions*.

*Alice in Wonderland*—as the tale has come to be known—and its sequel, *Through the Looking-Glass*, were written by Lewis Carroll, the pen name of Charles Lutwidge Dodgson, a mathematical logician at Oxford.

Because of the ascent of non-Euclidean geometry, and the attempts to find a firm foundation for arithmetic in set theory, the world of mathematics had turned its attention largely to logic in hopes that an analysis of the nature of mathematical reasoning would yield the needed justification to keep mathematics as the hardcore basis of all that was certain.

But mathematicians are a strange lot. While they understood the gravity of the circumstances, they also found themselves drawn to the paradoxes that could be created when these foundations were examined creatively.

The heart of traditional logic is the law of the excluded middle—the claim that either a sentence or its negation, but not both, must be true. Either I have a brother or I don’t; I can’t both have a brother and not have a brother.

If we know that one claim is true, we know the other is false. And if we know one is false, then we know the other is true. A paradox is a sentence or set of sentences that contradicts itself. That is, it must be true, but then its truth implies its falsity.

Since the law of the excluded middle holds that a sentence can’t be both true and false, we have an affront to the basis of logic itself. Logicians like Dodgson were examining purported paradoxes generated by the logical system itself. If authentic, such paradoxes would undermine the underpinnings of our most rigorous form of thought.

Learn more about math concepts explored by literary figures like Kurt Vonnegut

Testing paradoxes were not limited to Dodgson’s professional published work; it’s also what he was playing with in his famous work for children. Think of the opening scene in which Alice spies the White Rabbit: he takes a pocket watch out of his waistcoat, declares that he’ll be late, and then he dashes down a hole into which Alice follows.

Carroll famously wrote that “Either the well was very deep, or she fell very slowly, for she had plenty of time as she went down to look about her and wonder what was going to happen next.”

Now, remember that Dodgson was British and the most celebrated influential figure in Britain was Isaac Newton. Newton’s laws of motion explained Galileo’s finding that all objects close to the surface of the Earth fall at exactly the same rate.

If we’re taking seriously the possibility that Alice is falling slowly, then we’re taking seriously the possibility that the immutable laws we take as part and parcel of the working of the universe no longer apply.

We have entered a realm where the Enlightenment presupposition of a well-behaved universe whose rules are accessible to our rational faculties can be reasonably denied. Reason implies nonsense.

Reason is not ultimately self-justifying, but ultimately self-defeating. *Wonderland* represents the death of the rationalist project.

Think of Alice’s encounter with Humpty Dumpty in *Through the Looking-Glass*. The two are discussing birthdays and birthday presents when Humpty Dumpty exclaims, “There’s glory for you.” Alice replies, “I don’t know what you mean by glory.” To quote the passage further:

Humpty Dumpty smiled contemptuously. “Of course you don’t—till I tell you. I meant ‘there’s a nice knock-down argument for you!’”

“But ‘glory’ doesn’t mean ‘a nice knock-down argument’,” Alice objected.

“When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.”

“The question is,” said Alice, “whether you can make words mean so many different things.”

“The question is,” said Humpty Dumpty, “which is to be master—that’s all.”

In Greek, a language that Dodgson spoke, the word for “word” is *logos*, which also means, in some contexts, logic. When he has Humpty Dumpty redefine the word “glory,” it’s not a random sense but rather a nice knock-down argument, that is, the goal of logic.

When Alice protests, Humpty Dumpty replies that the real question is who is to be master: humans or words? That is, which ought to be the measure of reality: our experience, our freedom, our lives, or logic? Should we be subservient to our reasoning as the Enlightenment-influenced rationalists would have?

Or should we take command over logic, words, and *logos*? If we follow logic, do we disappear down a rabbit-hole—something that seemed possible given the paradoxes that mathematical logic was generating?

Learn more about the universe as changing and unstable

A similar challenge is found in Abbott’s *Flatland*, which takes place on a two-dimensional plane. It’s a flat world populated by shapes, the narrator being a lowly square. In Flatland, the more sides you have, the higher your social position.

The square is visited by a sphere, a three-dimensional figure that appears to the square at first as a point, then a circle of increasing diameter, then a circle of decreasing diameter, and then a point—finally, a disembodied voice. The sphere tries without any initial success to convince the square of the existence of the third dimension until he finally flings the square out of his planar world into the space above it.

Upon returning to Flatland, the square becomes evangelical about convincing his fellow flatlanders about the existence of this third dimension that is upward not northward—a dimension they haven’t seen.

He’s arrested and charged with heresy by the high priest, and at his trial, he is asked to provide any evidence for the existence of this third dimension of which he so passionately speaks.

The square’s argument is mathematical. If we can take a point and move it, we get a line. If we take a line and move it parallel to itself, we get a plane. If we take that plane and move it parallel to itself, we get space.

The priest asks for physical, rather than mathematical, reasoning. The square can provide none, so the priest offers his argument: He asserts that there is no reason to think this mathematical talk is anything but trickery, with no relation to anything real.

The argument is compelling; all the while the reader knows it’s wrong. But it’s reflective of the strange results coming out of mathematics at the time. They threatened to undermine our comfortable certain basis for rationality.

But should they be accepted? Has reason led to nonsense, or is there a foundation for rational thought to be found in rational thought?

Learn more about the paradoxical subject of quantum mechanics

For more than a thousand years, we accepted Euclid’s axioms and postulates as true because they were self-evident, seemingly self-justifying. But at the dawn of the 20th century, mathematics—our most secure and definitive science—was in turmoil.

We were forced to re-examine the basis of what we thought reality would be like. Lewis Carroll’s Wonderland was about to be discovered here in our world, when the science of the 20th century forced us to reconsider reality itself.

Lewis Carroll, whose real name was Charles Dodgson, was indeed a mathematician. At the time, groundbreaking new mathematical concepts were coming out such as imaginary numbers. Dodgson, whose views on math were rather traditional, considered these new concepts to be absurd, and the world of *Alice’s Adventures in Wonderland* mirrors this absurdity.

The scientific name for Alice in Wonderland syndrome is dysmetropsia, which is a neuropsychological condition in which one’s perception is distorted. In Lewis Carroll’s novel, Alice experiences a distorted sense of her body and physical surroundings, which reflects the anxiety many people were experiencing at that time as new mathematical concepts were introduced which changed how they viewed the world.

In the book *Flatland*, Flatland is a world occupied by flat, two-dimensional objects (circles, squares, etc.). Space, on the other hand, contains three-dimensional shapes such as spheres and cubes. The objects in Flatland can’t comprehend that this three-dimensional world exists because all they know is their own world.

Flatland and Lineland are similar in that each land consists of lines and points. Unlike the world of Spaceland, Lineland is familiar to Flatland because they share common elements.

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The early 20th century sat at an interesting intellectual crossroads. The Enlightenment of the 17th and 18th centuries upheld reason as the defining characteristic of humanity and saw the advance of knowledge as the hallmark of human progress.

The romantic movement of the 19th century was a backlash against what it saw as the arrogance and naiveté of the reduction of the human to its brain. As such, there was a divide in the intellectual world between the sciences and the arts.

This is a transcript from the video seriesRedefining Reality: The Intellectual Implications of Modern Science. Watch it now, on The Great Courses Plus.

The sciences largely bought into the Enlightenment presuppositions of a well-behaved world, regulated by absolute laws that were accessible to human reason through rigorous processes of observation and logic. Acquiring an understanding of these laws was paramount in striving to move forward as a species.

The practitioners of the arts and letters, on the other hand, saw themselves as the loyal opposition, obligated to correct what they saw as the overreach of the sciences, which seemed to miss the beauty, the joy, and the experience of being human. The heart was as important as the brain and the fetish that scientists held for knowledge limited their true understanding.

But this divide was not impermeable; there were important influences in both directions.

The advances achieved and the difficulties experienced by the sciences changed the way people saw the universe, the world, and human nature. This change in perspective affected what was painted, built, written, and composed. Art reflects the world, but the world is never given to us directly.

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As the late-18th-century philosopher Immanuel Kant pointed out, “Our understanding of reality is always mediated through concepts we use to create the ideas in our mind that only seem to come fully formed from our senses.”

But contrary to Kant, who held that these basic categories were necessary and unrevisable, these intellectual building blocks do change over time. With advances in the sciences, it forces us to radically revise how we make sense of ourselves and our environment.

This revision provides fertile ground for the creative arts. The freedom creatives enjoy to reflect the world in novel and sometimes strange ways can inform and influence the scientists, who are often in need of new and exciting ways to organize the seemingly strange results they receive from the universe.

Sometimes art and science influence each other, sometimes not. A tension existed between the Enlightenment-influenced rationalists and the romantically-inclined thinkers.

Reason-based rationalists supported their foundational views by citing the progress science and technology had made as evidence. The use of reason gave us justified beliefs that could come from nowhere else.

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Foremost amongst these were the propositions of mathematics, which provided humanity with absolute truths.

Rene Descartes—a 17th-century founder of this rationalistic movement, and a major contributor to physics, mathematics, and philosophy—thought that the methods of the mathematician were so impressive that they ought to form the backbone of all further investigations. In all other areas of conversation, the intellectuals disagreed about everything.

But mathematics demanded universal assent by way of facts that could not be challenged by anyone who understood them, and complex results were derived with absolute rigor.

Mathematics was a thing of beauty, an absolute bedrock on which man could construct a completely firm structure of understanding. A generation after Descartes, when Isaac Newton mathematized physics with his invention of calculus, it seemed like the rational worldview based on mathematics was well on its way to giving us an unassailable sense of reality itself.

Mathematical propositions were self-evident and true beyond question. Those who doubted these propositions revealed themselves either to be lacking in understanding, mentally deficient, or just trying to be irascible.

It was worrisome when, in the 19th century, the very foundations of mathematics came into serious doubt.

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Traditionally, the mathematical realm has been thought of as having two parts: Geometry, which deals with shapes in space, and arithmetic, which deals with matters of number.

Both had been rigorously grounded. While interesting works were showing some interconnections, it was thought there were two different, but equally justified areas of knowledge. Then things fell apart in both.

Since the 3rd century B.C., geometry was synonymous with the name Euclid. Centuries of work had been achieved in geometry before Euclid, but his contribution created order from the results.

He created a structure based on a few simple and obvious propositions using a strict means of reasoning to derive hundreds of complex and intricate theorems. These theorems, because of the rigor of his logic, must share in the certainty attributed to the first, most basic truths.

These basic truths come in three groups. First, are the definitions that simply describe what is meant by basic geometric terms.

A circle, for example, is the set of points in a plane some distance from a center point.

Definitions are true. They’re true because they simply tell us what we mean by words. We’re free to define any word in any way we want.

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But there are two other categories of basic truths Euclid used. One category includes his collection of axioms. The axioms were basic obvious truths that were not explicitly geometric.

For example, equals added to equals yields equals. If John and Suzy have the same number of apples and each is given some additional number of apples—giving them the same—then each still has the same number of apples as the other. It’s difficult to argue against that truth.

The postulates are similar except they are about purely geometric matters. Give us any two points and we can draw a line between them. Give us any line segment and we can continue that line as far as anyone wants in either direction.

Give a point and you can draw a circle around it any size you want. All right angles are equal to each other. No one could doubt these.

These are the first four of the postulates. Now, if the first four are fingers on the Euclidean hand, the fifth is the sore thumb—it sticks out.

This is the theorem: if two lines are approaching each other, they’ll eventually intersect. We usually think of it in terms of an equivalent formulation—take a line and a point not on that line.

How many lines can be drawn through the point that will be parallel with the line? One and only one.

This postulate seemed less like the others and more similar to Euclid’s theorems, the statements he proved from the other postulates.

Maybe it would be possible to derive it from the other four. This would be a big deal because mathematicians prize elegance. A system is elegant if it makes the fewest possible assumptions.

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To show how we could derive the fifth postulate from the other four, making it unnecessary as an assumption, it would shrink the set of presuppositions. This would improve Euclid’s system, the seemingly greatest, most elegant, and powerful system of all time.

The significance of improving upon Euclid’s ideas would be as great as improving upon the works of Shakespeare; it would assure one’s place in the annals of mathematical history. Much time was spent by brilliant people for centuries seeking the elusive proof of the fifth postulate—the so-called parallel postulate.

Such a proof was never found, presumably because it doesn’t exist. Euclid cannot be improved on, as mathematicians had hoped.

The fifth postulate is entirely independent of the other four, but mathematicians discovered this the hard way. After mathematicians had failed in all their attempts to create a direct proof from the first four to the parallel postulate, the idea occurred to several different mathematicians to try an indirect proof.

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We can show something is true by demonstrating that it can’t be false. If you know that I have a sibling and you want to prove that I have a brother, it suffices to prove that I can’t have a sister. If I have a sibling and it’s false that I have a sister, then it must be true that I have a brother.

What the mathematicians wanted to prove is that Euclid’s fifth postulate can be derived from the other four; that means that the truth of the other four postulates guarantees the truth of the fifth. We start by assuming the opposite: The other four postulates are true and the fifth is false.

Then we derive a contradiction by forming any sentence of the form “A and not A.” Since either A or not A has to be true, but both can’t be, the contradiction A and not A has to be false.

The existence of this contradiction shows that if the postulates one, two, three, and four are held to be true, then the denial of the fifth can’t be true. But if the denial of the fifth is false, then the fifth has to be true. This would show that Euclid could be simplified.

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However, when mathematicians assumed one, two, three, four, and the negation of five, they worked and worked but never found a contradiction. They found strange results, such as the discovery that triangles cannot have the same angles but different sizes; the internal angles of triangles add to less than 180 degrees.

Bizarre stuff—statements that seemed like they were false, but never a contradiction that had to be false.

The world was entering a new mathematical realm.

Rationalism is the theory that there are absolute truths which, using reason, the intellect can discover.

Within the tenets of rationalism, there is no proof or evidence of a supernatural creature that created and rules the universe.

It is generally accepted that René Descartes, the French philosopher, is the creator of rationalism.

Empiricism and rationalism are not the same. They are very nearly opposite. Empiricism denies innate truths and is the belief in the strict use of our five senses and induction to discover any truth, whereas rationalism believes there are innate truths that can be deduced with reason and intellect.

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For centuries, mathematicians had tried to simplify Euclid’s system—one of the most elegant systems in mathematics, and the basis for the geometry taught in schools today. To accomplish this feat, they would have to show how they could derive the fifth postulate from the other four.

Not only was a solution never found, but they made some bizarre discoveries along the way.

This is a transcript from the video seriesRedefining Reality: The Intellectual Implications of Modern Science. Watch it now, on The Great Courses Plus.

In the first half of the 19th century, mathematicians like Nikolay Lobachevsky of Russia realized that they had found something incredibly deep and troubling. They had in their hands a new geometry, a different geometry—a non-Euclidean geometry.

This strange world was a new mathematical realm—a parallel mathematical universe. If we have two geometries, which one is true? When we had only Euclid’s, we assumed that it gave us the absolute truth about the nature of shape and space.

If there’s a possible alternative, we can’t hold its truth to be absolute. We need a new sort of evidence to justify our belief in what seemed indubitable.

But, what kind of evidence could this be? We can’t simply say that the alternatives are too weird—being weird doesn’t make it false.

To make matters worse, more systems were created by denying other postulates and combinations of postulates. Possible geometric systems were popping up right and left.

Which one was true? Which one was the real geometry? How do we know? What had been the most secure place on the entire intellectual landscape for more than a thousand years was now suddenly without a foundation.

Learn more about the shocking discoveries of non-Euclidean geometries

Mathematicians were not happy, but at least we had the other side of the mathematical house. Arithmetic was still safe and secure; one plus one is two. There can’t be any reason to doubt that.

We had thought the numbers were well behaved, that they obeyed certain undeniable first truths. Think back to Euclid’s first axiom: Equals added to equals yields equals.

But let’s now think of the fifth axiom—the whole is greater than the parts. If I have an amount of money and in my will, I leave some of it to a relative and the rest to a charity, neither heir gets as much as I previously had in sum; by getting part, both get less than I had in aggregate.

This seems trivial and obvious. Of course, it’s always true.

In the second half of the 19th century, the German mathematician Georg Cantor showed this is not the case. Suppose we have a mutual friend named George and next week is George’s birthday.

We want to get him something we know he’ll love as a gift, but what to get him? You remind me that he’s an avid collector of numbers. We talk to his wife and she tells us that his collection now includes all of the positive integers.

He has 1, 2, 3, 816, 9,674,217—he’s got them all. If he has all the numbers you could count from 1 forward, we’ll also get him the one before 1—we’ll give him 0.

The big day comes and after blowing out his candle, he opens his gift, and sees his new 0, a number he didn’t have before. Overjoyed he looks at us and says, “Thanks for nothing.”

George’s number collection now has one more than it had before his birthday. The pre-birthday collection is only a part of the whole, so the whole is larger than the part, right?

Wrong; he still has the same number of numbers.

Learn more about the underlying reality that governs the universe

Suppose we go to a movie theater and we want to see if the show is sold out, undersold, or oversold. Since all people have but one backside and we use that backside to sit in but one seat, we could count the number of seats in the theater, count the number of backsides, and see which number is bigger or if they’re equal.

But Cantor realized we could do it in a way without counting at all. Ask everyone to sit down—that is, match every available seat to an available person, one to one. Then see if there is a remaining seat, a remaining individual, or neither.

This is a way to compare the size of sets without counting. Two sets are of equal size if there exists a way to map the members of one set onto the members of the other so that each element in the first set corresponds to one, and only one, member of the second with none left over.

Let’s do this with George’s numbers: If we take each of the numbers in his post-birthday collection, can we map it to one, and onto only one, from his pre-birthday collection? Take each number in the collection after his birthday and map it onto that number plus 1 in his old collection.

That is, 0 goes onto 1, 1 goes onto 2, 2 goes onto 3, and so on. In this way, in the end, there will be no number in one that does not have a correlated number in the other.

This maps one set perfectly onto the other set. The two collections are the same size.

George’s extra number has made his collection no larger even though it now includes a new element above and beyond what it had. For an infinite number—the number of counting numbers—that infinite number plus one yields the same number.

All right, that’s weird but we might think, “It’s all because it’s an infinite number of numbers. You can’t make infinity bigger. It is already infinite. How can you make it bigger?”

But all we showed is that infinite amounts are as big as one can get. You can’t have a smaller or larger infinity. This is where Cantor’s work gets fun.

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Consider the numbers between zero and one. Some of these are what we call rational numbers, that is, they can be written as ratios: one half is ½, three-quarters is ¾. These can be written equivalently as decimals: one half is 0.5, three-quarters is 0.75.

Interestingly, all of these numbers will have one of two properties. Either they will terminate, that is like ½ will end, 0.5, done. Or, they will repeat—one-third is 0.3333333 and as far you go, there will always be more threes.

One-seventh, when written as a decimal, is 0.142857142857142857, repeats infinitely. There will always be another 142857. Such is the case with all ratios: they terminate or they repeat.

Then there are the numbers that do not repeat or terminate when we write them out as decimals. These are what we call the irrational numbers—not because they’re crazy, but because they can’t be written as a ratio of two counting numbers.

The most famous irrational number, of course, is pi—3.14159 and off it goes forever, always another digit, never repeating endlessly like the 3’s of 1/3 or the 142857’s of 1/7. If you take the rational numbers and combine them with the irrational numbers, you get what we call the real numbers.

Suppose George has an older brother, Frank, who has been collecting numbers even longer. He’s amassed all the rational numbers between 0 and 1: ½, ¼, 9/16, all of them.

Frank then orders from an online retailer the set of irrational numbers between 0 and 1. When it arrives, he now has both the infinite set of rational numbers between 0 and 1 and the infinite set of irrational numbers between 0 and 1. That is, he has all of the real numbers between 0 and 1.

We might think that, like George on his birthday, Frank’s new set with more numbers is the same size as it was before. Infinity is infinity; you can’t have more than infinity.

But you would be wrong. Cantor proved with absolute certainty that Frank’s new set is a bigger infinity. There are sizes of infinity.

If we map Frank’s old set onto his new set, Cantor showed that there’s a simple way to demonstrate that the new set has at least one number that can’t be in the old set, which means the new set is bigger.

Some infinite subsets are the same size as the sets that include them, and other infinite sets are bigger than others. There would be an infinite set of infinite numbers and these obey different rules than the finite numbers.

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Which number rules are true? The easy answer would be that there’s one set of rules for finite numbers and another set for infinite ones.

That conclusion was the line mathematicians pursued until 1931 when the Austrian mathematician Kurt Gödel proved that we could not set out a complete set of rules for arithmetic.

Gödel showed that we could use any set of possible rules to create sentences similar to the sentence, “This sentence is false.” If it is true, then it is false, but if it is false, then it is true.

Any attempt to create rules would either allow sentences like, “This sentence is unprovable,” to be proven and so we would have sentences that can be proved but are false. We would have just proven the sentence that says it cannot be proven.

Alternatively, we could strengthen our rules to exclude these sentences, but then because we can no longer prove the sentence, the sentence “This sentence is unprovable,” would be true.

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We would have true sentences that we can’t prove in our system, making our system incomplete. Any set of rules would be either unsound—that is, include false sentences—or incomplete—not allow all true sentences to be proved.

The days of mathematics as the epitome of human rational understanding seemed to close at the end of the 19th and beginning of the 20th century. It was the canary in the intellectual coal mine.

An absolute truth is a concept or idea that is true no matter what, such as the rule that a circle can never be square.

There are absolute truths in mathematics such that the axioms they are based on remain true. Euclidean mathematics falls apart in non-Euclidean space and different dimensions result in changes. One could say that within certain jurisdictions of mathematics there are absolute truths.

Mathematics was not invented. The Kemetic priests of Egypt taught a wholistic concept of number and sound which became a cult led and taught by Pythagoras to the Greeks. Around 300 B.C.E., the axiomatic system we still use to discover mathematical insights was developed by Euclid.

Mathematics appears to exist as a part of this universe. There is a mathematical universe hypothesis by Max Tegmark that posits that the universe itself is a mathematical structure. It is possible in other universes that we could not understand them and they would not be mathematical; however, our perception within this universe is mathematical and so even considering these other possibilities is difficult.

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There is an inspiring article that highlights thinking outside of the box when it comes to solving problems. The article compares a gifted class of middle school students with a remedial sort of vocational class, the article was measuring the creativity of the students. The experiment asked the students in each class one question: *How do you weigh a giraffe?*

The students in the gifted class were not so much gifted as successful, and they were used to succeeding and used to pleasing their teacher. They panicked because they didn’t know how to answer this question. This was way before the Internet, and the students couldn’t go online and look it up.

Meanwhile, in the vocational class, almost immediately some kid just blurted out and said, “Hey, I know what to do. Just take a chainsaw, and chainsaw that giraffe into chunks. Then weigh the chunks.”

Chainsawing the giraffe is an attitude that a good problem solver should have because you want to be fun, and you also want to be a little bit bad. Breaking rules is a good thing when we’re not talking about actual cruelty to animals here. We’re just talking about thinking outside the box or breaking mathematical rules.

To break the mathematical rules, we need a healthy dose of the 3 C’s.

This is a transcript from the video seriesArt and Craft of Mathematical Problem Solving. Watch it now, on The Great Courses Plus.

Concentration, creativity, and confidence are psychological attributes that are important for just about everything, but they’re vital for solving problems. How do we enhance them? All three of them are linked, but confidence is the least important of the three because it’s truly derived from the other two.

As your concentration ability increases, and if your creativity gets stronger, then you’ll naturally become more confident.

Learn more about thinking like a problem solver

To master concentration, you must set aside a quiet time and place for your work. You need to relax. You need to develop good work habits, and you need to find problems to concentrate on that are interesting to you, approachable by you, and addictive. Pretty much, that will do the trick. In order to build up your concentration, you want to build up from level 1, which is a minute or so of concentration, at least to level 3, getting up to an hour.

Collect a stock of back-burner problems. Start cultivating problems that you cannot solve. Make sure they’re interesting and then you’ll think about them. If you can find a problem that’s exciting to you, that annoys you, that sort of gnaws at you, then you’ll think about it. Interesting problems will force you to become a better concentrator.

*To master concentration, you must set aside a quiet time and place for your work.*

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Concentration leads to confidence, which frees you to explore, which facilitates investigation and creativity.

To build both your confidence and creativity, you need to be disciplined about using those interesting problems. You need to set up a problem-solving routine, some workplace, a lucky pen, and then you should keep to your routine to get your mind in a relaxed state.

Then, occasionally, deliberately break your routine. If you like to work in the morning, work late at night. If you like quiet, go to a noisy café. If you like to work in a restaurant, go sit in a library, etc.

You should also, as a strategy, specifically think about peripheral vision. The peripheral vision strategy is to realize that many problems cannot be solved with direct focus. It’s just like your eyes. Your fovea has very good focus, but it has less sensitivity than the perimeter of your eyes, the periphery of your vision.

*Peripheral vision strategy—many problems cannot be solved with direct focus. *

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Many problems need to percolate in your unconscious in this way. You need to cultivate a good supply of back burners, and just get in the habit of not solving problems. The more you do this, the more you’ll get into a state of investigative, purposeful contemplation, and the more powerful your mind will get.

Learn more about the psychological aspects of problem solving

Let’s look at a tool made famous by Carl Gauss. He was a prodigy, and as a teenager, he solved a problem that had been unsolved since Hellenistic times. He found a way to construct a regular heptadecagon, 17-gon, using compass and straightedge. The rest of his career was not much different. What Gauss could do in an afternoon was equivalent to what an ordinary mathematician could do in a lifetime.

When he was 10, he was faced with the problem: How do you find the sum of the numbers 1 + 2 + 3 up to 100? How do you compute this in 1787 when there are no calculators? Well, what little Gauss did was to pair the terms, the beginning term and the end term, (1 + 100); and then the second term and the next to last term, (2 + 99); and then (3 + 98); (4 + 97); and so on down to (50 + 51). Each of those pairs adds up to 101, and there are 50 such pairs. Thus, the sum is 5050, and that’s pretty clever. This is called Gaussian pairing and is an example of a powerful and useful tool.

Learn more about the power of specific tools, or “tricks”, to make a mathematical expression simpler

Wishful thinking is one of your first strategies for this because pretending to solve a problem, even an easier one, keeps you happy. It allows you to keep thinking about solving problems. Even delusion helps – deluding yourself into thinking that you’ve solved a problem actually allows you to solve it later because you can relax and be happy. Making yourself happy and confident, even if it’s through such a transparent thing as delusion, is fine.

A corollary of wishful thinking is a very sensible idea, which I just call the “make it easier” strategy. The idea is completely common sense. If your problem is too hard, just make it easier by removing the hard part. Either make the size smaller or remove an element that makes it hard. For example, if it involves square roots, remove them temporarily.

What you should keep in mind is strategy and tactic is what makes someone a good problem solver, not the tools. Now, if you’ve never seen the Gaussian pairing tool, which Gauss used to sum the numbers from 1 to a 100, you are undoubtedly impressed. Gaussian pairing are quite clever, but they are tools and tools are just tricks. These are things that can be acquired. What you should keep in mind is that strategy and tactic are what makes someone a good problem solver, not the tools.

And you should use these new ideas. Any time you see a new, interesting idea, learn it, use it, and make it yours. Ideas are collective human property. They are not private property. Don’t forget that what you’re doing is chainsawing the giraffe. It’s okay to mess around and break some rules.

Learn more about three strategies for achieving a problem-solving breakthrough

Let’s look at a quickie: a problem that requires “think outside the box” thinking.

If you consider the problem of nine dots in a grid and ask how do you join them all by drawing no more than four absolutely straight lines? If you think outside the box , as demonstrated in the diagram, it’s pretty obvious what to do.

As long as you go outside the box, you’re able to get all 9 dots. It’s a fun and challenging problem if you’ve never seen it before.

*Thinking outside the box helps you become a good problem solver. *

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People may be endowed unequally with confidence, creativity, and power of concentration, but all of these are trainable skills. It’s possible to practice them and improve them, but in order to do so, you will need to see lots of creativity in action and you need lots of open-ended opportunity to experiment.

Learn more about the hidden world of problem solvers

The seven steps of problem solving are: identify a problem, define goals, brainstorm, consider alternatives, agree on the solution, execute the solution, and evaluate the outcome.

Good problem solvers generally have less drama in their lives and react less emotionally to difficulties and thus are able to follow the steps of problem-solving more coherently.

Become a better problem solver by stimulating the brain with mathematical problems and games, removing drama from your life, and following the steps of problem solving while allowing for random clues to appear.

Solving problems gets better with experience and team leaders must lead by offering solutions and roads to solutions which arise from experience.

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Let’s say you buy a lottery ticket; what are the chances that you’re going to be rich for the rest of your life?

You walk across a golf course in a stormy day; what are the chances you’ll be hit by lightning?

What are the chances that your investments will allow you to live happily for the rest of your days?

You have a fever; you have a cough. What are the chances that it’s a serious disease rather than something trivial?

This is a transcript from the video seriesWhat Are The Chances? Probability Made Clear.Watch it now, on The Great Courses Plus.

All these are real-life examples of situations where we’re confronted with possibilities whose outcomes we do not know. In fact, I would argue that many or most parts of our lives—and the world and trying to understand the world—involve situations where we don’t know what’s going to happen. They involve the uncertain and the unknown.

Learn more about the concept of randomness and its quantification through probability

It would be nice to say, “Well, our challenge in life is to get rid of uncertainty and be in complete control of everything.” That is not going to happen. One of life’s real challenges is to deal with the uncertain and the unknown in some sort of an effective way; and that is the realm of probability.

Probability accomplishes the really amazing feat of giving a meaningful numerical description of things that we admit we do not know, of the uncertain and the unknown. It gives us information that we actually can act on.

For example, when we hear there’s an 80% chance of rain, what do we do? We take an umbrella. Of course, if it doesn’t rain, we say, “Well, there was a 20% chance it wouldn’t rain. That’s okay.” If it rains, we say, “Oh, yes, the prediction was right. There was an 80% chance of rain.”

Probability is a rather subtle kind of a concept because it can come out one way or the other, and still a probabilistic prediction can be viewed as correct—but decisions made on probability have all sorts of ramifications.

When we make medical decisions, for example, we are making decisions that are based on probabilities, and yet they have extremely serious consequences, including life and death consequences.

In the case of the rain, all we risk is getting wet. But in many areas of making decisions on the basis of probability, there are very serious consequences. When we make medical decisions, for example, we are making decisions that are based on probabilities, and yet they have extremely serious consequences, including life and death consequences.

Learn more about a numerical way to make decisions

Back before probability was viewed as commonplace as it is today— between 1750 and 1770 in Paris, there was a smallpox epidemic for which a vaccine was developed. Unfortunately, the inoculations were rather risky. They reckoned that there was a 1 in 200 chance of death from taking the inoculation, but on the other hand, there was a 1 in 7 chance of dying eventually from the disease. So making that kind of decision is a very dramatic question where we’re weighing probabilities.

If you took that inoculation and you died immediately from smallpox, did you make the right decision or not? Well, of course, you don’t want to be among the 1 in 200 that died from the inoculation. On the other hand, on the basis of probability, it was the right decision. There are many controversies about this kind of thing and in today’s world with lawsuits and all this would be a very serious kind of an issue to undertake.

Well, in many arenas of life, our understanding of the world comes down to understanding processes and outcomes that are probabilistic in nature, that really come about from random chance, that things are happening by randomness alone. Over the last couple of centuries, the scientific descriptions of our world increasingly have included probabilistic components in them.

Learn more about probability can be used to model the distribution of genetic traits

Many aspects of physics all involve questions of probability. Things we imagine—molecules causing things to happen by the aggregate force of probabilistic occurrences—quantum mechanics — thermodynamics. At the very foundations of our knowledge of these studies is the theory of probability.

Biology, genetics, and evolution are all based very centrally on random behavior, as well. In fact, in all of these areas, the goal is to make definite, predictable, measurable statements about what’s going to happen that are the result of describing random behavior.

The description of random behavior is how we, as scientists and mathematicians, define the world. This is a major paradigm shift in the way science has worked for the last 150 years. As time goes on, there continues to be an increase in the role of probability and randomness at the center of scientific descriptions.

Probability gives us a specific statement about what to expect when things happen at random. But how can it be effective when, by definition, random outcomes of one trial or one experiment are completely unknown? Well, if you repeat those trials many, many times and look at them in the aggregate, that’s when you begin to see glimpses of regularity. It’s the job of probability to put a meaningful numerical value on the things that we admit we don’t know.

Probability is a mathematical possibility of what might occur when one takes part in a mathematical potential such as rolling dice or choosing an item off a menu.

There are many types of probability: experimental, theoretical and subjective probabilities are the most commonly understood.

A coin toss is the most common example of probability. With a 50 percent chance of either heads or tails, we see both sides of the issue.

Simple probability is simply the probability of an outcome such as a roll of the dice or choosing an item from a menu.

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When used in statistics, the word population refers to the entirety of the collection of people or things that are of interest. A sample is a subset of the total population.

In general, the goal is to infer information about the whole population from information about the sample. In other words, it’s not in our interest to know only about the people who are asked in the sample. What we’re really interested in is those aspects of the entire population.

This is a transcript from the video seriesMeaning from Data: Statistics Made Clear.Watch it now, on The Great Courses Plus.

If you choose the sample randomly, the advantage is that using probability you can make inferences about how well the opinions of the sample do, in fact, represent the opinions of the whole population.

On the other hand, if you intentionally choose certain groups to reflect what you believe to be reflective of reality, you may bring your own biases to the selection process, and those biases are then going to be reflected in the people whom you ask. Representative of the whole population means that the sample should have the same characteristics that the whole population does.

The whole concept of choosing the sample randomly is that you have a better chance that the proportion of people in the sample with a certain opinion will be, in fact, the same as the entire population.

Learn more about induction within polling and scientific reasoning

The most familiar occasion where this comes up is before an election, when pollsters try to find out what proportion of the voters will vote for the Democratic candidate and what proportion will vote for the Republican candidate.

There are several major pitfalls in the way sampling can be done. In the 1936 U.S. presidential election, the two primary contenders for the presidency were the incumbent, Franklin Delano Roosevelt, and the Republican opponent, Alfred Landon. At the time, the magazine *The* *Literary Digest* had for several elections conducted polls to predict who would win the coming election. They had successfully predicted the outcomes in several elections, so this was a major poll.

In the 1936 election, *The Literary Digest* sent out 10 million voting surveys, and they received 2.4 million replies. Based on those surveys, *The* *Literary Digest* predicted that Landon would win in a landslide, with 370 electoral votes to Roosevelt’s 161.

Well, you may not recall reading about President Landon in your American history books. Obviously he did not win the presidency.

In fact, the only correct aspect of *The* *Literary Digest*’s prediction was that the election was a landslide, but unfortunately for them, the landslide was the other way. Roosevelt won the election with 62 percent of the popular vote and by an incredible 523 electoral votes to 8 for Landon.

Learn more about what makes aggregation more effective than any single poll

Obviously, *The* *Literary Digest*’s sampling method was not representative of the whole population.

What went wrong? Well, one thing was that *The* *Literary Digest* got their samples from several different kinds of lists. One list was the subscribers to their own magazine. They also looked at car registration records, and that was an available list of a lot of names, and they sent their surveys to those people. They also used telephones.

The people to whom

TheLiterary Digesthad sent their survey were likely wealthy people and obviously their opinions were not representative of the population at large.

The people to whom *The* *Literary Digest* had sent their survey were likely wealthy people and obviously their opinions were not representative of the population at large.

The year 1936 was in the middle of the Great Depression, and many people were having financial problems and were cutting back on their budgets. Probably one of the first things to go in tight times would be one’s subscription to *The* *Literary Digest*. In addition, not many people owned cars or telephones. These were luxury items for a lot of people in 1936. Because of this, the people to whom *The* *Literary Digest* had sent their survey were likely wealthy people and obviously their opinions were not representative of the population at large.

Learn more about gathering data from which deductions can be drawn confidently

*The* *Literary Digest* poll’s second pitfall was that it was a voluntary response survey.

The magazine sent out all these surveys, and only some people replied. The problem with this is that sometimes people who send back replies have a particular bias. Instead of sending back replies in the same proportion, maybe some people with a certain opinion are more apt to reply. The bias that can come from voluntary responses may not just give an answer that’s a little off, but it can give a completely erroneous view of reality.

Because of this story, *The* *Literary Digest*, which otherwise would simply be lost in the dustbin of history, will now live on forever in statistics textbooks as a great example of bias in sampling.

A success that came from this *Literary Digest* fiasco is the story of George Gallup.

At the time, Gallup was a young statistician just starting out, and he did his own poll for the 1936 election. He took a survey of 50,000 people and made two predictions of his own for the election.

- He correctly predicted that Roosevelt would win the election.
- He also predicted that
*The**Literary Digest*poll would be wrong and estimated how wrong they would be before their poll came out.

He was one of the people who introduced the concept of randomness in political polling as a key feature of sampling techniques. That is absolutely one of the fundamental criteria to look for when you’re evaluating whether a sample survey is, in fact, a good one.

Learn more about sampling; a technique for inferring features of a whole population from information about some of its members

Randomness is a basic ingredient of essentially all of the standard statistical techniques, and the reason it’s an ingredient is because the analysis of randomness and probability that allow us to apply mathematics to the understanding of the results that we get.

The most basic way to get an accurate sample is to take a sample that’s called a simple random sample, which is, as the name implies, simply to take the entire population you’re interested in, and say how many people you want to survey and randomly select them from that group, and then get the answer from each member of that selected sample.

Of course, there are lots of problems in getting the answer from that selected sample. But the simple random sample is the gold standard for finding a representative sample.

Political polling is used both to predict a campaign’s results and to give a candidate or supporters of that candidate a metric by which to measure the candidate’s results. The first poll, called a benchmark, helps a candidate to design a campaign strategy by identifying such factors as the candidate’s overall popularity, the demographics of the people most likely to vote for that candidate, and the issues that matter most to the candidate’s main audience.

An exit poll is conducted after an election. As voters leave the polling station, reporters ask them who they voted for. This is used to predict election results, since the votes can sometimes take a few days to count.

Polling data is used by a political candidate to gather information about how well he/she is resonating with potential voters at the start of and during a campaign. It provides the candidate with information such as the demographical features of the individuals who would most likely vote for the candidate and allow the candidate to test the popularity of various messages. His/her overall approval rating is also a good indicator of whether or not it is worth staying in the campaign because running a political campaign is very expensive.

Push polls are intended to “push” an issue to the forefront of the voter’s mind. For example, a push poll might ask potential voters to evaluate candidates based on their support of healthcare.

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While the origins of π are not known for certain, we know that the Babylonians approximated π in base 60 around 1800 B.C.E. The definition of π centers around circles. It’s the ratio of the circumference of a circle to its diameter—a number just a little bit bigger than three.

The constant π helps us understand our universe with greater clarity. The definition of π inspired a new notion of the measurement of angles, a new unit of measurement. This important angle measure is known as “radian measure” and gave rise to many important insights in our physical world. As for π itself, Johann Lambert showed in 1761 that π is an irrational number, and later, in 1882, Ferdinand von Lindemann proved that π is not a solution to any polynomial equation with integers. However, many questions about π remain unanswered.

Learn More: Geometry—Polygons and Circles

Any discussion of the origins of pi must begin with an experiment involving circles that we can all try. Take any circle at all and take the length of the circumference—which is the length around—and measure it in terms of the diameter, which is the length across. You will end up with three diameters and just a little bit more, and if you look closely, it’s a little bit more than 1/10 of the way extra. This experiment shows us that that ratio of the circumference to the diameter is going to be a number that’s around, or a little bit bigger than, 3.1. No matter what the size of the circle is, the circumference is slightly greater than three times its diameter.

This is a transcript from the video seriesZero to Infinity. Watch it now, on The Great Courses.

This fixed, constant value was given a name, and we call it π. How do we say it more precisely? The number π is defined to equal the ratio of the circumference of any circle to its diameter across. This ratio is constant. No matter what size of the circle we try this with, that number will be always the same. It begins 3.141592653589, and it keeps going.

The symbol π comes from the Greek letter π, because the Greek word for “periphery” begins with the Greek letter π. The periphery of a circle was the precursor to the perimeter of a circle, which today we call circumference. The symbol π first appears in William Jones’s 1709 text *A New Introduction to Mathematics*, and the symbol was later made popular by the great 18th-century Swiss mathematician Leonhard Euler around 1737.

Learn More: Number Theory—Prime Numbers and Divisors

Moving from its name to its value, evidence exists that the Babylonians approximated π in base 60 around 1800 B.C.E. In fact, they believed that π = 25/8, or 3.125—an amazing approximation for so early in human history. The ancient Egyptian scribe Ahmes, who is associated with the famous Rhind Papyrus, offered the approximation 256/81, which works out to be 3.16049. Again, we see an impressive approximation to this constant. There’s even an implicit value of π given in the Bible. In 1 Kings 7:23, a round basin is said to have 30-cubit circumference and 10-cubit diameter. Thus, in the Bible, it implicitly states that π equals 3 (30/10).

The Indian mathematician and astronomer Aryabhata approximated π, in c. 500 C.E., with the fraction 62,832/20,000, which is 3.1416—a truly amazing estimate.

Not surprisingly, as humankind’s understanding of numbers evolved, so did its ability to better understand and thus estimate π itself. In the year 263, the Chinese mathematician Liu Hui believed that π = 3.141014.

Approximately 200 years later, the Indian mathematician and astronomer Aryabhata approximated π with the fraction 62,832/20,000, which is 3.1416—a truly amazing estimate. Around 1400, the Persian astronomer Kashani computed π correctly to 16 digits.

Let’s break away from this historical hunt for the digits of π and consider π as an important number in our universe. Given π’s connection with measuring circumferences of circles, scholars were inspired to use it as a measure of angle distance. Consider a circle having radius 1. Radius is just the measure from the center out to the side. It’s half the diameter.

The traditional units for measures of angles are, of course, degrees. With degrees, one complete rotation around the circle has a measure of 360 degrees, which happens to approximately equal the number of days in one complete year and which might explain why we think of once around as 360.

Instead of the arbitrary measure of 360 to mean once around the circle, let’s figure out the actual length of traveling around this particular circle, a circle of radius 1, once around. What’s the length and the circumference of that? If we have a radius of 1, then our diameter is twice that, 2, and so we know that the once-around will be 2 times π because the circumference is π times the diameter.

Once around will be 2π. One full rotation around, which is an angle of 360 degrees, would be swept out with circumference length of 2π in this particular circle. Halfway around would be 180 degrees, and we would sweep out half of the circumference, which, in this case, would be π. Ninety degrees would sweep out a quarter of the circle, and for this particular circle, that would have length π/2, or one-half π.

We’re beginning to see that every angle corresponds to a distance measured part- or all the way around this particular circle of radius 1. In other words, for any angle, we can measure the length of the arc of this circle swept out by that angle.

This arc length provides a new way of representing the measure of an angle, and we call this measure of angles “radian measure.” For example, 360 degrees = 2π radians, those are the units; 180 degrees equals π radians, and 90 degrees would equal π/2 radians. Remember, all these measures are always based on a special circle that has radius 1.

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It turns out that this radian measure is much more useful in measuring angles for mathematics and physics than the more familiar degree measure. This fact is unsurprising. Radian measure is naturally connected through the circumference length with the angle, rather than the more arbitrary degree measure that has no mathematical underpinnings. It represents an approximation through a complete year.

The term radian first appeared in print in the 1870s, but by that time, great mathematicians, including the great mathematician Leonhard Euler, had been using angles measured in radians for over a hundred years.

The number π appears in countless important formulas and theories, including the Heisenberg uncertainty principle and Einstein’s field equation from general relativity. It’s an important formula and number across the world.

Many equations represent Pi in its entirety, but as it is an irrational number, its decimal representation beginning with 3.14159… keeps going forever, at least when calculated.

There are many ways to calculate Pi, but the standard method is to measure the circumference of a circle with string or tape, measure the diameter with a ruler, and divide the circumference by the diameter. Pi = Circumference / Diameter.

It is not known whether Pi can end; there is only theory, which so far, cannot prove or disprove Pi ending or being infinite.

Technically, no one invented Pi. It was always there as a ratio of a circle’s circumference to its diameter. It is known to have been calculated as far back as ancient Sumer, and the Rhind Papyrus from ancient Egypt shows Pi calculated to 3.1605.

This article was updated on April 28, 2020

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