Chapter 33 - The case of i
Chapter 33
The i example
All through the day, I me mine, I me mine, I me mine
All through the night, I me mine, I me mine, I me mine
Now they're frightened of leaving it
Everyone's weaving it
Going on strong all the time, All through the day, I me mine.
Wikipedia says:” "I Me Mine" is a song by the English rock band the Beatles from their 1970 album Let It Be. Written by George Harrison, it was the last new track the group recorded before their break-up in April 1970.”
George wouldn’t like it, but here’s a different take:
All through the day, we know we exist
All through the night, we know we exist
(etc - you can make up the rest).
And we can cheekily ascribe Harrison’s *i sing therefore I am* to Rene Descartes, renowned as the father of modern philosophy, renowned for his aphorism “I think therefore I am” (= cogito ergo sum), and *also* renowned for advancing his signature mind-body dualism.
But Descartes was also a mathematician, and one of the funniest ideas he came across was the notion that there could be square roots to negative numbers.
Now, ask any junior high school student with an interest in maths and you will be advised in no uncertain terms: *there is no* square root to a negative number.
Here we go:
2x2=4, 3x3=9, correct?
And -2x-2=4, -3x-3=9 as well, correct?
So what is the square root of -9?
*There is none*. Not from minus infinity to positive infinity, not in the tiny numbers close either side to zero, there is no number anywhere along the number line that can be squared to produce a negative outcome. In any real sense, the square root of a negative number does not exist.
That’s why Descartes thought the idea ludicrous. Remember, too, that Descartes had no problem believing in the God of the Bible, and went to some pains to attempt to “prove” the existence of God in his “Discourses and Meditations”. As a Catholic of his time he may well have believed in literal transsubstantiation, … but he didn’t believe in any square root to a negative number.
So, why deride something impossible in the first place?
Because certain Italian mathematicians had imported from Arabian scholars, in the 12th century, a notion that it could be useful to invent a square root to a negative number.
By the early 16th century (yep, took some time) the solutions to certain cubic equations (what are they? Doesn’t matter here) had been published on the basis of imagined square roots to negative numbers.
And Descartes called these square roots “imaginary”
He coined the term imaginary in his 1537 book La Geometrie:
“For the rest, neither the false nor the true roots are always real, sometimes they are only imaginary, that is to say one may imagine as many as I said in each equation, but sometimes there exists no quantity corresponding to those one imagines.”
and
“For any equation one can imagine as many roots [as its degree would suggest], but in many cases no quantity exists which corresponds to what one imagines.”
And what is the point here? The point is that, of all the sciences, mathematics is held to be the most logical, the most pure, the most pitilessly rational. Mathematics is not supposed to just make stuff up out of thin air.
But ask any *senior* advanced math high school student, or any university math undergraduate, or any engineering undergraduate, or virtually anyone with a degree in quantitative science, and they will confirm - they use these *imaginary* numbers as a matter of course. Not because they *exist*, but because *pretending* these imaginary numbers exist helps solve *real* problems that *do* exist.
Shadows of an Imaginary: “i” and the Ontology of Scientific Constructs
I. Introduction: The Square Root of Minus One
There are few symbols in the history of mathematics more curious, more paradoxical, and more indispensable than the humble i — the square root of –1. René Descartes, in La Géométrie (1637), first dubbed such quantities imaginary, and the name stuck. For centuries, that label carried a pejorative sting: something fanciful, perhaps useful for bookkeeping, but not “real.” And yet, by the twentieth century, i had become the beating heart of modern physics and engineering. Alternating current (AC) circuits, quantum mechanics, control theory, Fourier analysis, signal processing — all depend on complex numbers. Remove i, and the lights go out, computers collapse, and quantum theory evaporates.
The irony is obvious: a number once ridiculed as fictitious has proven to be more indispensable to human survival than most “real” numbers. And this irony makes it a perfect test case for philosophy of science. For what, after all, is required for a construct to be accepted into the canon of human knowledge? Must it be observable? Must it be falsifiable? Must it be part of a fruitful research programme? Or is “usefulness” enough?
This chapter explores i as a philosophical object. We will trace its historical emergence, apply the major twentieth-century criteria of scientific legitimacy to it (logical positivism, Popperian falsificationism, Kuhnian paradigms, Lakatosian programmes, Quine–Putnam indispensability, van Fraassen’s constructive empiricism, Hacking’s entity realism), and then broaden the frame to ask: what does the status of i tell us about the ontology of many other constructs in science? Quarks, wavefunctions, dark matter, even “fitness” in evolutionary theory — all share a similar half-light of reality. They are “darkly certain”: never directly graspable, always mediated, yet indispensable.
In the end, I will argue that science is not — and cannot be — a pure mirror of metaphysical reality. It is, rather, a pragmatic web of constructs. And its constructs are accepted not because they meet some transcendental criterion, but because they work.
II. A Brief History of the Imaginary
Descartes and the “Imaginary”
René Descartes’ 1637 La Géométrie introduced the word imaginary (imaginaire) to describe the square roots of negative numbers. Descartes dismissed them: “These numbers are called imaginary because they exist only in the imagination” (La Géométrie, Book III). For him, they were algebraic curiosities without real referents.
Bombelli’s Courage
But the seeds were already sown earlier. Rafael Bombelli (1526–1572), in his Algebra (1572), had attempted to make sense of such numbers while solving cubic equations. He wrote:
“It will appear to you an extravagant thing that I should make use of this kind of quantity, which, in fact, is merely imaginary. For all that, we shall see that operations with it will prove consistent.”
Bombelli’s genius was methodological: he treated “imaginary” numbers as if they were legitimate, followed the algebra, and discovered consistent results.
Euler, Gauss, and Legitimacy
Euler (1707–1783) took the next leap. He introduced the symbol i for √–1, and in 1748 wrote the extraordinary identity:
eiπ+1=0eiπ+1=0
a single equation uniting the exponential, the imaginary, the circular, the multiplicative, and the additive — mathematics’ crown jewel.
Gauss (1777–1855) finally legitimized complex numbers. In Theoria residuorum biquadraticorum (1831), Gauss argued they should be regarded as points on a plane — what we now call the Argand–Gauss plane. No longer “imaginary,” they were simply “lateral” extensions of the number line.
Riemann, Hilbert, Bourbaki
By the late nineteenth century, Riemann’s complex analysis had revealed the extraordinary fertility of i in physics and mathematics. Hilbert and the Bourbaki group in the twentieth century further normalized abstraction: mathematical objects need not be “real” in any naive sense; consistency and fruitfulness were sufficient.
Thus, by 1900, the “imaginary” had become one of the most “real” of numbers — at least in the sense of intellectual indispensability.
III. Philosophical Frameworks: What Makes a Construct Acceptable?
To evaluate i through the lens of philosophy of science, we must consider the major schools of the twentieth and twenty-first centuries.
Logical Positivism
The Vienna Circle (Carnap, Schlick, Ayer) sought to eliminate metaphysics by tying meaning to verification. A statement is meaningful if it can be translated into observation language. On such a criterion, i seems disqualified: no observation corresponds to √–1. One cannot “measure” it.
But positivists already had to admit mathematics as a “tautological” system — meaningful not by observation, but by its role as analytic framework. As Carnap wrote in The Logical Structure of the World (1928):
“Mathematics is analytic; its statements add nothing factual, but provide the form into which factual statements may be put.”
Thus, for the positivists, i is acceptable — not because it refers, but because it structures.
Popper: Falsifiability
Karl Popper, in The Logic of Scientific Discovery (1934), proposed falsifiability as the demarcation of science. Theories are scientific if they can be refuted. Can i be falsified? Not directly. But theories deploying it can. Quantum mechanics, electrical engineering, control theory — all depend on i, and these have been rigorously tested. If they had failed, i would have been implicated.
Popper would thus regard i as acceptable via its role in falsifiable theories.
Kuhn: Paradigms
Thomas Kuhn’s The Structure of Scientific Revolutions (1962) would interpret complex numbers as a paradigm shift within mathematics: from the number line to the number plane. Once adopted, i restructured “normal mathematics.” Its success was not in correspondence to reality but in generating coherent puzzle-solving within a new framework.
Kuhn’s analysis underscores the sociological aspect: acceptance came not because i was “true,” but because a community found it fruitful.
Lakatos: Research Programmes
Imre Lakatos, in The Methodology of Scientific Research Programmes (1978), argued that the rationality of science lies in whether a programme is “progressive” (predicting novel facts) or “degenerative” (ad hoc patching). Complex analysis was spectacularly progressive: it yielded elegant proofs, new theorems, and indispensable physical applications. Thus, iwas justified.
Quine and Putnam: Indispensability
The Quine–Putnam indispensability argument is decisive:
“We must believe in the existence of all entities that are indispensable to our best scientific theories.”
Since quantum mechanics, electromagnetism, and modern computing are unintelligible without i, we must (on pain of scientific incoherence) accept its existence.
Van Fraassen: Constructive Empiricism
Bas van Fraassen’s The Scientific Image (1980) argued we need only believe in the empirical adequacy of science, not the reality of its posits. From this view, ineed not exist; it suffices that the theories using it predict correctly.
Hacking: Entity Realism
Ian Hacking’s Representing and Intervening (1983) proposed entity realism: if you can manipulate a concept to intervene in the world, it is real enough. Engineers manipulate i daily in phasor diagrams to run power grids. Thus, i is as real as a voltmeter needle.
IV. Does i Meet the Criteria?
Applying these frameworks:
● Positivists: i is analytic structure.
● Popper: i earns its keep in falsifiable theories.
● Kuhn: i was a paradigm shift.
● Lakatos: i belongs to a progressive programme.
● Quine–Putnam: i is indispensable.
● Van Fraassen: i is empirically adequate.
● Hacking: i is manipulable, hence real.
Every philosophy, even those hostile to metaphysics, finds a rationale to accept i.
V. Parallels in the Physical Sciences
The case of i is not unique. Science is filled with constructs that live in ontological twilight.
● The electron. Never “seen,” only inferred from tracks, fields, or images. J. J. Thomson (1897) spoke of “corpuscles”; Feynman later quipped: “The electron is an idea which has no place to hide.”
● Quarks. Proposed by Gell-Mann (1964), but never isolated. Richard Feynman called them “mathematical conveniences” before reluctantly accepting their indirect evidence.
● The wavefunction ψ. Central to quantum mechanics, but is it a real field or a probability tool? Einstein: “The wavefunction does not describe the state of one system, but the state of our knowledge.”
● Dark matter and dark energy. Inferred from galactic rotation curves and cosmological expansion. Unseen, yet accepted.
● Fields. Once mysterious, now standard. Faraday thought of them as “lines of force,” Einstein as geometry of spacetime.
All these resemble i: unseen, abstract, yet indispensable.
VI. The Utility Criterion
William James’s pragmatism offers the plainest answer:
“Truth is what works.” (Pragmatism, 1907).
John Dewey, too: inquiry is “instrumental.” Wigner’s famous essay, The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960), marvelled that entities like i not only work but work uncannily well.
Putnam, in Mathematics, Matter and Method (1975):
“It would be a miracle if our scientific theories were so successful if they were not at least approximately true.”
VII. Epistemic Humility
Scientists themselves have long recognised provisionality. Newton (Opticks, 1704): “Hypotheses non fingo” — “I feign no hypotheses.” Osler (1902): “Half of what you are taught will be wrong.”
The electron may be reconceptualised; spacetime may be replaced. But like i, constructs earn their place by utility, not ontology.
VIII. Conclusion: Married to the Imaginary
The history of i shows that human knowledge cannot be judged only by correspondence to metaphysical “reality.” i does not “exist” in the same way as a stone, yet it governs the machines that power our cities.
So too with electrons, quarks, wavefunctions, and dark matter. Science is not embarrassed by this. It is married to the imaginary, not because scientists mistake shadows for reality, but because the imaginary works.
Thus, despite their humility, scientists proceed with constructs about which they are only “darkly certain.” They persist not from metaphysical arrogance but from pragmatic confidence. And in this confidence lies the true epistemic strength of science: not to mirror reality perfectly, but to build bridges, predict eclipses, cure diseases, and — yes — eradicate smallpox.
As Churchill once said of democracy, that it is the worst system of government except for all the others, so too of science: it is the worst way of knowing truth — except for all the others. And i stands as its emblem: imaginary, yet indispensable, fictional, yet foundational, unreal, yet the most real of all.
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