Norman Wildberger: The Problem with Infinity in
Math
Summary | Theories of Everything Podcast (Curt Jaimungal) | Chronological Condensation
Introduction
Curt Jaimungal interviews Norman Wildberger, professor of pure mathematics at the University of New
South Wales and founder of the YouTube channel "Insights into Mathematics." Wildberger is described
as "one of the rare examples of a mathematician examining the foundations of mathematics in order to
reformulate it, ridding it of the qualities considered abhorrent by the intuitionists." The discussion
centers on whether the concept of infinity is well founded and the "reality or unreality" of real numbers.
Overview of Wildberger's Position
Wildberger clarifies he doesn't have a "grandiose revision of mathematics" but rather a way of thinking
that is "more concrete and explicit" than most practitioners. He sees benefits in adopting a more
restrictive, careful, computationally-based approach. His projects include algebraic calculus, rational
trigonometry, hyperbolic geometry, and recent work on solving polynomial equations.
The Problem with Infinity
Wildberger's main reservation concerns assuming we can "do an infinite number of things." He
distinguishes between ongoing infinity (an unbounded process that remains finite at any given moment)
and completed infinity (treating infinite collections as objects available for further construction). The
ancient Greeks, including Euclid, viewed lines as extendable but finite at any point—never as infinite
objects. Aristotle discussed this dichotomy. Modern 20th-century mathematics replaced this classical
respect for infinity's unbounded nature with treating completed infinities as prior objects, which
Wildberger considers "a kind of arrogance."
"At the heart of my objection is this implicit belief that people have that they can do something that
they actually demonstrably cannot do."
The Computational Litmus Test
Rather than philosophical debate, Wildberger proposes a concrete test: can a computer encode the
mathematical objects we claim to work with? Whether working with infinite decimals, Dedekind cuts,
Cauchy sequences, or infinite sets underlying manifolds and topological vector spaces, we assume
collections we don't actually possess. When we write the symbol ℝ, that's just a symbol—not the real
numbers themselves. We need to adopt a "more modest and sober position" and recognize what we can
and cannot do.
Algorithmic Reality
Asked if he believes underlying reality is computational like Wolfram suggests, Wildberger says he
doesn't hold that opinion, though it may end up being close to what's happening. He feels "we're not
close enough to the bottom yet" to make confident assessments. However, he does believe the
algorithmic approach to mathematics is logically solid—providing "a surety that transcends arguments
just with phrases and words."
Demonstration with Exponential Functions
Wildberger demonstrates the problem using Taylor series. The exponential function e
x
is defined as an
infinite sum. Even truncating at 10
10
10
terms—far short of infinity—the computation already
"overwhelms the computational aspects of the entire universe." At 10
10
10
10
terms, this becomes certain.
Since we cannot calculate even this finite sum, "we have no right to assert the existence or say anything
really much about a sum which is vastly superior, namely going to infinity."
Definitional vs. Existence Claims
Wildberger reframes the issue: it's not primarily about existence but definition. "What is the definition of
infinity or e
7
?" The definition of e
7
as an infinite sum is invalid because we cannot reach the end. "We
cannot get to the end of the rainbow. So for us to say, 'let G be the pot of gold at the end of the rainbow'
is not a proper definition."
The Circle and Exact vs. Approximate
When asked whether treating impossible objects as computational conveniences (like architects using
idealized circles) is acceptable, Wildberger notes the circle is historically central to these difficulties.
Ancient astronomers creating trig tables knew exact values were impossible—this was necessarily
approximate. He emphasizes the distinction between intrinsically exact and intrinsically approximate
mathematics. Adding distinctions complicates theory but makes it "more accurate and representative of
reality."
Real Number Arithmetic Doesn't Work
Wildberger demonstrates that computers cannot perform exact real number arithmetic. Asking a
computer to evaluate π + e + √2 returns the same expression unevaluated. The same occurs with log(4) +
tan(7) or cos(3) × sin(7). Numerical approximations work fine, but exact arithmetic on "very simple real
numbers" with "very simple operations" produces no calculation. Meanwhile, rational arithmetic like 1/2
+ 1/3 + 1/5 = 31/30 works perfectly.
The Infinite Decimal Problem
Analysts knew before Dedekind that infinite decimals were problematic. Adding two infinite decimals
requires starting from the rightmost digit, but infinite decimals have no rightmost digit. The "carry of
nines" problem means you can never guarantee the n
th
decimal place is correct. Dedekind cuts and
Cauchy sequences were attempts to "circumvent these difficulties, not face them squarely." This creates
mathematics "walking around on stilts, not looking down"—separated from computational reality.
Physics Unaffected
Wildberger explains why physicists like Ed Witten don't concern themselves with these issues: to
physicists, each decimal digit is one-tenth the previous in importance. By the 30th digit, "we are no
longer interested." Questions about whether decimals go to infinity are "completely irrelevant" to
physics—they truncate immediately to finite approximations. Applied mathematics goes "from strength
to strength." It's pure mathematics that has "the serious problems."
Babylonian Mathematics
Wildberger's interest in ancient mathematics includes the Babylonian base-60 system, which was "in
many ways more powerful" than ours. Because 60 = 2 × 3 × 5, fractions with denominators containing
only these factors have finite sexagesimal representations. The set of "regular numbers" is vastly larger
than in base-10. If starting mathematics from scratch, "there would be a good argument to be made to
use the Babylonian base 60."
Complex Analysis Preserved
Asked whether the beauty of complex analysis is removed by his approach, Wildberger says he doesn't
think so—though it requires redoing traditional theories. Analytic functions play a larger role in complex
analysis than real analysis. The essential beauty will "shine through" and perhaps become clearer
because "the essential aspect of it will become clearer."
Why No Contradictions?
If mainstream mathematics is flawed, why hasn't it produced major inconsistencies? Wildberger explains
the problems are "problems of ambiguity," not logical contradictions. We've assumed things that
"obscure the fundamental nature of our reality." The consequence is confusion and "missing out on a
better and more accurate and more powerful way of thinking."
The Intermediate Value Theorem
Asked how to rephrase the intermediate value theorem without real numbers, Wildberger connects this to
the fundamental theorem of algebra. The reformulation must state things in terms of approximate zeros,
intervals, and accompanying numerical analysis—purely in rational number terms avoiding mention of
infinite processes. This is "a major major to-do in pure mathematics."
Not Finitism
Wildberger clarifies he's not a finitist—he doesn't believe there's necessarily a biggest natural number.
His metaphor: "There's a road and I can walk down this road and pick up an integer and walk back. If I
want a bigger integer, I go further." There's no end to the road, but also no "big ballpark in which
everything is happening." He's "more of a classical Greek thinker"—recognizing the unbounded aspect
while remaining "agnostic as to what's down that road beyond my view."
"I can use any natural number without claiming to use every natural number."
Philosophical Position
When asked if he's an idealist or Platonist, Wildberger says categories aren't as helpful as attributes like
being "honest, clear, and careful." He does believe there's a mathematical aspect to reality—sharing his
colleague James Franklin's Aristotelian view that "mathematics actually can be found here in the world."
But there may be many possible points of view on that mathematical reality.
Document Type: Summary – Condensed while preserving chronological conversation flow
Source: "Norman Wildberger: The Problem with Infinity in Math" – Theories of Everything Podcast
Original Length: ~8,000 words | Summary Length: ~1,400 words (17%)