The Problem with Infinity in Mathematics
Norman Wildberger's Constructivist Critique and Alternative Foundations
Digest | Source: Theories of Everything Podcast (Curt Jaimungal) | Guest: Norman Wildberger, Professor of Pure
Mathematics, UNSW
Bottom Line Up Front
Core Thesis: Modern mathematics rests on an unjustified assumption—that we can perform
infinite operations—which leads to foundational objects (real numbers, infinite sets) that
cannot be computed or defined in any meaningful sense. This is not a philosophical objection
but a computational one: what we cannot encode on a computer, we cannot legitimately claim
to work with.
Key Distinction: The ancient Greek view of "ongoing" infinity (always extendable, never
completed) was replaced by 20th-century "completed" infinity (treated as an actual object).
Wildberger argues this replacement constitutes an arrogance—claiming we can do what we
demonstrably cannot.
Practical Impact: Applied mathematics and physics remain largely unaffected since they use
finite approximations. The problems lie in pure mathematics, where foundational ambiguities
propagate through manifolds, varieties, topological vector spaces, and Lie groups.
The Central Problem: Definition Before Existence
Wildberger frames his critique not as an existence claim but as a definitional one. Before asking "Does
infinity exist?" or "Does e
7
exist?", we must ask: what is the definition of these objects? His central
claim: these definitions fail because they presuppose completing infinite processes.
The Ongoing vs. Completed Infinity Distinction
Ongoing infinity (Classical Greek view): Objects are always finite but extendable. A line in
Euclid can be extended indefinitely, but at any moment it remains a finite object. Natural numbers
can be counted further, but we always possess only a finite range. This view respects that infinity
"goes beyond our view."
Completed infinity (Modern view): Infinite collections are treated as prior objects available for
further construction. We write ℝ and claim to possess "the real numbers" as an actual collection.
Wildberger calls this replacement a loss of "long standing respect" for infinity's unbounded nature.
The Computational Litmus Test
Rather than engaging in philosophical argument-trading, Wildberger proposes a concrete test: Can a
computer encode and embody the mathematical objects we claim to work with? This shifts the
discussion from metaphysics to demonstrable capability. When we write "e
7
," we write a symbol, not the
number—and our computers cannot compute what that symbol allegedly represents.
"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 Failure of Real Number Arithmetic
Wildberger demonstrates that real number arithmetic—the foundation of modern geometry and analysis
—cannot actually be performed. This is not about practical limitations but fundamental impossibility.
Demonstration: π + e + √2
Ask any computer algebra system to compute
π + e + √2
. It returns the same expression
unevaluated. Similarly for
log(4) + tan(7)
or
cos(3) × sin(7)
. While numerical
approximations are trivial, exact arithmetic on "very simple real numbers" performing "very
simple arithmetical operations" yields no calculation. The computer—and therefore we—are
incapable of real number arithmetic.
Contrast with rational arithmetic:
1/2 + 1/3 + 1/5 = 31/30
. A primary school calculation
that computers perform exactly.
The Infinite Decimal Problem
The historical prototype of real numbers—infinite decimals—faces a fundamental algorithmic
impossibility. To add two infinite decimals, you must start from the rightmost digit and carry leftward.
But infinite decimals have no rightmost digit. Any attempt to truncate, calculate, and extend faces the
"carry of nines" problem: the n
th
decimal place might depend on operations arbitrarily far down the
sequence. You can never be certain you have the correct digit.
Dedekind cuts and Cauchy sequence equivalence classes were attempts to "circumvent these difficulties,
not face them squarely." They create a theory that walks "on stilts so that we never touch the floor"—
mathematics separated from computational reality.
The Scale Argument: Beyond Computational Capacity
Even before reaching infinity, we encounter computational impossibilities. Consider the exponential
function's Taylor series truncated at 10
10
10
terms—vastly short of infinity but already exceeding the
universe's computational capacity even if we wrote at Planck scale.
The Universe as Computational Limit
A sum from k=0 to 10
10
10
10
of 1/k! × x
k
cannot be computed by the entire universe. Yet we claim
confident knowledge about the "exact" infinite sum. Wildberger: "We have no right to assert the
existence or say anything really much about a sum which is vastly superior, vastly bigger than this
one, namely going to infinity."
Empirical Evidence: Atrophy Through Tool Reliance
The interview surfaces neurological evidence paralleling Wildberger's mathematical concerns about
outsourcing to abstractions we cannot ground.
GPS and Spatial Cognition
London taxi drivers who memorized city layouts showed physically larger brain regions for
spatial knowledge. Drivers who switched to GPS saw measurable gray matter shrinkage. The tool
that "does the work for us" atrophies the capability it replaces.
The Pure vs. Applied Mathematics Split
Wildberger makes a critical distinction: his objections target pure mathematics, not applied mathematics
or physics.
Why Physicists Are Unaffected
Every decimal digit is one-tenth the size of the previous. By the 30th digit, physicists are no longer
interested—it doesn't affect calculations. They immediately truncate e and π to finite decimal
approximations. Questions about whether decimals "go on to infinity" are "completely irrelevant" to
physics. Wildberger "completely sees" why Ed Witten doesn't concern himself with foundational issues.
Why Pure Mathematics Has Problems
Pure mathematics claims exact, not approximate, objects. The "exactness that they currently ascribe to
something like the exponential function or the cosine or the sine... that exactness is illusory." This
creates "confusion at various levels and missing out on a better and more accurate and more powerful
way of thinking."
"Applied mathematics is going from strength to strength. It's pure mathematics that has the
serious problems."
Not Finitism: The Road Metaphor
Wildberger carefully distinguishes his position from finitism—he does not claim there is a largest natural
number.
The Road vs. The Basket
Common misconception: If we can't use "every" natural number, there must be a maximum.
Wildberger's view: "There's a road and I can walk down this road and pick up an integer that's
there and walk back. If I want a bigger integer, I go further down the road." There's no end to the
road, but also no "big ballpark in which everything is happening." We can use any natural number
(pick one, I'll find the corresponding object) without claiming to use every natural number (as if
they're all in a basket we can manipulate).
He identifies as "more of a classical Greek thinker"—recognizing the road's unbounded aspect
while remaining "agnostic as to what's down that road beyond my view."
Alternative Foundations and Projects
Rational Trigonometry
Traditional trigonometry depends on transcendental functions (sine, cosine) with intrinsically
approximate values—even ancient astronomers knew exact tables were impossible. Rational
trigonometry works over general fields with general quadratic forms, unifying Euclidean and relativistic
geometries. It makes the "essential aspects clearer" rather than removing beauty.
Algebraic Calculus
Wildberger's project to redo calculus with a "polynomial orientation"—understanding the polynomial
story as stepping stone to the analytic story, grounding infinite series in algebraic structure rather than
completed limits.
Historical Number Systems
The Babylonian base-60 system was "in many ways more powerful" than base-10. 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, making their decimal arithmetic "demonstrably
more powerful."
Constructive Implications
Replacement theorems needed: The Intermediate Value Theorem and Fundamental Theorem
of Algebra must be restated in terms of approximate zeros, intervals, and accompanying
numerical analysis—purely in terms of rational numbers, avoiding infinite processes.
Extension fields for radicals: If you want √2 or ∛7 exactly, construct them algebraically as
extension fields rather than assuming real number existence.
Distinguish exact from approximate: Make sharp the distinction between intrinsically exact
and intrinsically approximate mathematics—complicating theory but increasing accuracy.
Computer science should have more say: Foundational issues need input from those who
understand computational constraints—people like Stephen Wolfram "have a very deep feeling
for foundations of mathematics."
Why No Contradictions Have Emerged
If mainstream mathematics with real numbers is "so flawed," why hasn't it collapsed?
Wildberger's Response
The problems are not contradictions but ambiguities. We haven't assumed something that leads to
logical contradiction—we've assumed things that "obscure the fundamental nature of our reality."
The consequence is confusion, not collapse. By facing the music and accepting harder, more
careful, more elementary thinking, "we can discover so many new interesting things and perhaps
help the physicists."
Philosophical Position: Aristotelian, Not Platonist
When asked if he's an idealist or Platonist, Wildberger deflects categories in favor of attributes: "honest,
clear, and careful." But he endorses his colleague James Franklin's Aristotelian view: "Mathematics
actually can be found here in the world." He's "happy to believe there's some mathematical aspect to
reality" but emphasizes that human-centered study admits "very many possible points of view" on that
mathematical reality.
Document Type: Digest – Thematically reorganized with synthesis for conceptual clarity
Source: "Norman Wildberger: The Problem with Infinity in Math" – Theories of Everything Podcast
Original Length: ~8,000 words | Digest Length: ~2,000 words (25%)
Related Resources: Norman Wildberger's YouTube channel "Insights into Mathematics"; Rational Trigonometry;
Algebraic Calculus course