Veritable Number System
| Scientific Theory | |
|---|---|
| Name | Veritable Number System |
| Type | Proposed number system |
| Author(s) | Peter F Erickson |
| Keywords | square root of minus one, imaginary numbers, negative numbers, direction, sign incompatibility, quaternions, vectors, infinitesimals |
| Year | 1999 (first circulated); 2005 (first published paper) |
The veritable number system is a number system proposed by the American independent scholar Peter F Erickson, which he claims to have discovered inside the square root of negative one. On his account it stands alongside the real numbers and the absolute numbers as one of three distinct ways of handling the concept of direction — not as an extension of the reals, and not as the imaginary axis of the complex plane.
Erickson first circulated the proposal in 1999 and published it in 2005 as The Solution to the Mystery of the Square Root of Minus One, developing it across five further papers and the book The Nature of Negative Numbers (2011).
The problem: what is √−1?
The system begins from an old discomfort. The square root of minus one has no interpretation as a magnitude, and mathematics has offered two ways of living with that. On the first, it is something transcendental — a quantity of a different order, admitted because it works. On the second, favoured by the logical positivists, it is not a quantity at all but merely a direction to perform an operation, a piece of notation with no referent.
Erickson rejects both, and notes what has been built on the unresolved question:
Most consider it to be either something transcendental or a mere direction to perform a certain operation. Einstein used it to help make plausible his idea of the fourth dimension. Actually, neither the idealists nor the logical positivists are correct. The −1 inside the radical sign is from a number system discovered by the author. It is called the "Veritable Number System." … It stands with the real number system and absolute numbers as one of the three ways to handle the concept of direction.
— Peter F. Erickson, The Solution to the Mystery of the Square Root of Minus One (2005)
His diagnosis is that the expression is not mysterious but mixed: the −1 beneath the radical does not belong to the real number system at all. It is a veritable number, and the appearance of paradox arises from applying an operation from one system to a quantity drawn from another.
The incompatibility of signs
The system's defining property is that positive and negative veritable numbers are incompatible with one another for the basic operations — addition, subtraction, multiplication, division, roots and ratios.
This is the substantive break with the reals. In ordinary arithmetic, sign is a property that can be freely combined away: −3 and +3 may be added to give zero, or multiplied to give −9. In the veritable system they cannot be so combined, because sign is not an attribute attached to a magnitude but a mark of direction belonging to the number itself. Two quantities pointing in opposite senses are not two values of one kind that happen to differ in sign; they are of different kinds.
Erickson argues that this is what makes the system better suited than the reals to describing paths, whether straight or curvilinear — because a path has a direction intrinsically, and a formalism that lets direction cancel loses information the situation actually contains.
Applications
- Quaternions and vectors. In Quaternions, Vectors, and Veritable Numbers (2009), Erickson examines the two systems that historically competed to handle directed quantities, and judges that neither got the matter right — neither Hamilton's quaternions nor the Gibbs vector calculus that displaced them. Both, on his view, work around the problem of direction rather than resolving it.
- Division. Division in the Veritable Number System (2011) argues that division is more versatile in the veritable system than among the reals — a claim connected to his wider contention that division by zero becomes valid under statable conditions once space is understood as composed of indivisible infinitesimals.
- Negative numbers. On Understanding Negative Numbers (2010), expanded into the book The Nature of Negative Numbers (2011), treats negativity itself as the phenomenon the veritable system exists to explain.
- Further development appears in Further Discoveries About the Veritable Number System (2009).
Relation to Erickson's wider position
The veritable numbers are not an isolated technical proposal. They belong to a single programme that also includes the spatial infinitesimal — the doctrine that space has a smallest, indivisible part and is not infinitely divisible — and the defence of absolute space, absolute time and absolute motion against Einstein.
The connection runs through Erickson's view of what mathematics is for. He holds that there is no difference in kind between mathematical and ordinary inductive reasoning, and that mathematics therefore has no special warrant to overrule intuition about space. A formalism that cannot be pictured — non-Euclidean geometry, the set-theoretic continuum, an imaginary axis with no referent — is on this view not a discovery about reality but a manipulation of symbols. The veritable system is his attempt to supply, for direction, the intelligible foundation he thinks the reals and the complex plane lack. See Peter F Erickson and Absolute Space, Absolute Time, & Absolute Motion.
Reception
The veritable number system has attracted essentially no engagement outside the Natural Philosophy Alliance, to which Erickson presented it across nine consecutive conferences. No mathematician appears to have published on it.
Reviewing The Nature of Negative Numbers, Kirkus found that "the author's ideas and reasoning are compelling", but judged the system an intellectual curiosity rather than a working tool, on the ground that mathematics and science rest on the ordinary laws of the real numbers. That assessment — coherent, interesting, unused — remains a fair summary.
The substantive objection is that the complex numbers are not in difficulty: they are consistent, complete in the algebraic sense, and indispensable across physics and engineering, so a replacement motivated by the interpretation of √−1 rather than by any failure of it faces a heavy burden. Erickson's answer is that consistency and utility are not the same as intelligibility, and that a system used successfully for centuries may still rest on a confusion about what its symbols denote.
Papers on this wiki
- 2011 – Division in the Veritable Number System
- 2010 – On Understanding Negative Numbers
- 2009 – Further Discoveries About the Veritable Number System
- 2009 – Quaternions, Vectors, and Veritable Numbers
- 2005 – The Solution to the Mystery of the Square Root of Minus One