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True Story: When AI Reinvents Mathematics
Mathematics doesn’t always begin with axioms. Sometimes, it begins with asking: “What if?”
Author: “What mathematical consequences would follow if we defined the absolute value
of a number to be negative - say, $|x|=-1$?”
ChatGPT (initial response): “That would be, frankly, not mathematics but pure fantasy - intriguing perhaps,
but outside the valid system.”
But the author wasn’t satisfied. He reminded the AI: something very similar had already
happened in the history of math, when mathematicians boldly defined $i := \sqrt{-1}$
and thereby created an entirely new number system: the complex numbers $\mathbb{C}$.
Rethinking the Impossible
Given that historical precedent, the author’s question became much more serious: Could we not do something similar for $|x|=-1$ by redefining the absolute value function itself? Could we invent a new number system, where such a definition made sense? The answer from ChatGPT this time: Yes - and here’s how.
Numbers in the SigAbs Space
Constructing a new number space, let’s imagine a two-dimensional vector space: $\mathbb{S} := \mathbb{R}^2$. We define a modified absolute value for any point $x = (x_1, x_2)$ in this space using a Minkowski-style metric: \[|x|_{\mathbb{S}} = \mathrm{SigAbs}(x) = \mathrm{sign}(x_1^2 - x_2^2) \cdot \sqrt{|x_1^2 - x_2^2|} \] This means: $|x|_{\mathbb{S}} = \begin{cases} \ \phantom{-}r & \text{if }\enspace x_1^2 > x_2^2 \\ \ \ \ 0 & \text{if }\enspace x_1^2 = x_2^2 \\ \ -r & \text{if }\enspace x_1^2 < x_2^2 \end{cases} \quad \text{with } r = \sqrt{|x_1^2 - x_2^2|}>0$.
This new absolute value behaves very differently from the Euclidean norm: It violates the classical triangle inequality and other familiar properties of norms, but $|x|_{\mathbb{S}} = -1$ is not paradoxical anymore.
From Idea to Visualization
What began as a mathematical thought experiment turned into a concrete model including a JavaScript plotter to visualize the structure of the space $\mathbb{R}^2_{\mathbb{S}}$. Over the next few days, the author struggled with ChatGPT’s:
- partial forgetfulness across sessions,
- inconsistent syntax hints,
- plotting issues with different versions of the Plotly JavaScript library.
Final Success
But persistence paid off. After two intense days of fighting code and logic - and additional help from DeekSeek - he had it: a visualization of the new space $\mathbb{S}$, complete with level curves of elements with equal absolute value, including those with negative modulus. The level curves are hyperbolas, rather than circles as in the Euclidean norm.
Watch the result at page 2.
AI reinvents mathematics