Now I Know in Part: Georg Cantor and The Limits of Human Reason

In 1895, the great mathematician and logician Georg Cantor published the first part of his defining work:

Contributions to the Founding of the Theory of Transfinite Numbers. In it, he solidified much of his career’s work by

studying the nature of infinity and fathering the field of set theory. He helped establish the foundations of

mathematical logic, along the way proving that there are an infinite number of infinities, distinguished by size.

For Cantor, the importance of his mathematics extended far beyond his academic career: his understanding

of infinity determined his understanding of the divine. During periods of discovery he wrote that ideas were

revealed to him by “higher energies.”1 In his 1895 publication, he included the following quote from Seneca’s

Naturales Quaestiones: “The time will come when those things which now lie hidden will be brought to light

by the day and the diligence of a future age.”2 Through Seneca’s words, Cantor envisioned the divine light of

reason being sustained by the glow of human ingenuity. Therefore, when his own faculties were insufficient

to reveal a way past the uncertainty of his conjectures, Cantor began to despair. He began questioning God.

In 1897, his contributions to mathematics essentially ceased when his efforts to win over the broader

scientific community fell short. Tormented by his inability to prove a conjecture called the Continuum

Hypothesis, Cantor seemed to feel humiliated by the opposition to his work levied by mathematicians,

philosophers, and even theologians. Simultaneously, his life’s work and his understanding of the world

seemed to be crumbling around him. Soon, he applied for a leave of absence from his university and

attempted to leave the study of mathematics altogether.

Cantor’s condition worsened when his son Felix died in 1899. He wrote to a friend expressing that his thirty-

year-long love for mathematics had vanished. For the next few years, he taught at the University of Halle as

much as he could manage. Then, in 1905, while attending a conference with his wife and daughters, he was

humiliated by a paper that claimed to disprove the Continuum Hypothesis entirely.3 As he watched the

upending of his life’s work, dismantling the very foundation of his worldview, his mind collapsed.

Although the 1905 paper was later debunked, restoring his reputation, his mental faculties never recovered.

Cantor spent the rest of his life in increasingly long periods of hospitalization—the final of which occurred in

1917. He died of a heart attack the next January at the age of 72.

To this day, the Continuum Hypothesis remains unproven. Later research on the hierarchy of logical systems

demonstrated that Zermelo-Frankel set theory, the framework which Cantor helped develop in order to

prove his most influential theorems, was unable to prove or disprove the Continuum Hypothesis. The

mathematical world his hypothesis inhabited was unable to prove an axiom upon which its consistency relied.

Later, Kurt Gödel and Paul Cohen proved its independence from Zermelo-Frankel set theory and believed

that it might be impossible to prove entirely. Since then, attempts to prove or disprove the Continuum

Hypothesis have shown more about the logical systems in which theorems are expressed than the theorem

itself.4 It is an illustrative example that no matter how high the human mind reasons, there will always be

truths that are beyond the limits of provable comprehension.

For my thoughts are not your thoughts,

neither are your ways my ways, declares the Lord.

For as the heavens are higher than the earth,

so are my ways higher than your ways

and my thoughts than your thoughts.

—Isaiah 55:8-9 (ESV)

An understanding of the limits of human reason can spare us from Cantor’s fate. In order to have any true

grounding in the world—especially in mathematics—we must recognize that objective truth exists. But we

must not expect the limited systems of reasoning our minds employ to fully comprehend the Truth to which

we aspire. Not only is it impossible for the human mind to know everything—it is impossible for the human

mind to know how to know everything.

This is the limitation conveyed in Isaiah 55. If we can’t truly store absolute truth in our minds, how can we

understand an absolute God? And if we can’t understand God, how can we hope to know Him or to love

Him? If the mind of God is above any logical system—both infinitely complex and perfectly sound—then

how can we hope to rise to the level of His thoughts in order to pursue Him? Even before our morality

enters the picture, we have no way of knowing God by our own power. Even if His attributes are apparent to

us, what our eyes see will not be understood fully in our minds.

This futility is a testament to the profundity of the incarnation. When He was born in a human body, Christ

stepped down from Heaven, not only in a physical sense, but also in a mental sense. And by being in perfect

communion with the Father, and making Himself known to men, Jesus allowed man to commune with the

Father in a way that we can never fully comprehend, as long as we live in an Earthly body. The futility of our

cognitive capacity necessitated that God acted first, of His own, incomprehensible, perfect love for us.

For now we see only a reflection as in a mirror, but then face to face. Now I know in part, but then I will

know fully, as I am fully known.

—1 Corinthians 13

Georg Cantor’s quotation of Seneca in 1895 has been misattributed to 1 Corinthians 13.5 However, the Spirit

through Paul gives us a hope that Seneca cannot. By knowing Christ as both man and God, our knowledge is

made complete in the hope of eventual unity with Him. Here, again, the example of Cantor serves as a

cautionary tale. After his hospitalization in 1905, Cantor wrote to an English colleague of an “inspiration

from above” which he experienced during his time of solitude. According to Cantor, this inspiration showed

him that Jesus of Nazareth was not born of a virgin, but was the son of Joseph of Arimathea. This apostatic

revelation shows us the tragedy of Cantor’s intellectual confusion. When Cantor lost his faith in the divinity

of his mathematics, he lost his faith in the divinity of Christ, and soon, his hope altogether.

Both crises of faith were underpinned by a desire for personal intellectual completeness. Yet an

understanding of our own incompleteness ought to motivate an even greater desire for completeness in

Christ. More radically and beautifully, we can rest in our incompleteness, knowing that, by faith, we can hold

fast to truths which are hard to reason about. For, “we know in part […] but when completeness comes, the

partial will pass away.”6 If our lives reflect this hope, then we have confidence that “He who began a good

work in you will bring it to completion in the day of Jesus Christ.”7 For it is not by knowledge but by faith

and steadfastness that we are made “perfect and complete, lacking nothing.”

Endnotes

1. Joseph W. Dauben, “Georg Cantor: The Personal Matrix of His Mathematics,” Isis 69, no. 4 (1978):

534-50, http://www.jstor.org/stable/231091.

2. Georg Cantor, “Beiträge Zur Begründung Der Transfiniten Mengenlehre,” Mathematische Annalen 46,

no. 4 (November 1895): 481-512, https://doi.org/10.1007/bf02124929.

3. Dauben, “Georg Cantor: The Personal Matrix of His Mathematics.”

4. Peter Koellner, “The Continuum Hypothesis.” Stanford Encyclopedia of Philosophy, May 22, 2013,

https://plato.stanford.edu/entries/continuum-hypothesis/.

5. “Dangerous Knowledge,” directed by David Malone, 2007, on BBC.

6. 1 Corinthians 13:10. ESV.

7. Philippians 1:6. ESV.

8. James 1:4. ESV