Rationalism in Mathematics Enters Shaky Ground

From the lecture series: Redefining Reality — The Intellectual Implications of Modern Science

By Steven Gimbel, Ph.D., Gettysburg College

During the Enlightenment, the world was viewed as an orderly place in which everything operated in accordance with precise mathematical principles. Beginning with the Romantic backlash, however, and unfolding into the early 20th century, both rationalism in mathematics and the concept of absolute truth were called into question. Discover how the quest to improve Euclid’s system of geometric propositions further disrupted the philosophy of rationalism.

Detail from Raphael's The School of Athens fresco (1511)
Detail from Raphael’s The School of Athens, featuring a Greek mathematician—perhaps representing Euclid or Archimedes—using a compass to draw a geometric construction. (Image: Raphael – Stitched together from vatican.va/Public Domain)

The Divide Between the Arts and the Sciences

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 series Redefining 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.

Learn more about how the scientific picture of reality changes as theories are refined or overthrown

Is Our Understanding of Reality Flexible?

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.”

Immanuel Kant - Picture from Meyers Lexicon books written in German language. Collection of 21 volumes published between 1905 and 1909
(Image: Nicku/Public domain)

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.

Learn more about revolutionary ideas about reality that eventually inspired the Age of Enlightenment

Mathematics as Absolute Truth

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

Rene Descartes (1596-1650). Engraved by W.Holl and published The Gallery Of Portraits With Memoirs encyclopedia, United Kingdom, 1833
French mathematician, physicist and philosopher Rene Descartes. (Image: Georgios Kollidas/Shutterstock)

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.

Learn more about how mathematics hit a patch of uncertainty in the 19th and 20th centuries

The Fundamentals of Geometry

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.

Euclid is an ancient Greek mathematician, the first theoretical treatise on mathematics
Since the 3rd century B.C., geometry was synonymous with the name Euclid, the Greek mathematician. (Image: German Vizulis/Shutterstock)

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.

Learn more about how Einstein’s special theory of relativity heralded an entirely new conception of reality

Characterizing Axioms and Postulates

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.

Learn more about the paradoxical subject of quantum mechanics

The Quest to Improve Euclid’s System

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.

Learn more about how quantum field theory led to a stunning synthesis called the standard model of particle physics

What is an Indirect Proof?

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.

Learn more about the underlying reality that governs the universe

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.

Common Questions About Rationalism in Mathematics

Q: What exactly is rationalism?

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

Q: Do rationalists accept God?

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

Q: Who created rationalism?

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

Q: Are empiricism and rationalism the same?

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.

This article was updated on November 21, 2019

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About Kate Findley 161 Articles
Kate is a writer, novelist, and blogger living in Los Angeles. She has been writing for The Great Courses since 2017. It incorporates her two favorite things: writing and learning.