In his recent Mind Matters News article, “What AI Has and Hasn’t Solved Recently in Math,” Robert J. Marks argues that recent advances in AI-driven mathematics exemplify horizontal rather than vertical innovation. While I agree with his distinction between these two forms of innovation, I believe the article attributes horizontal innovation to the wrong entity.

I agree with the article’s distinction between horizontal and vertical innovation. As the article mentions, vertical innovation occurs when an entirely new method is invented to solve a problem. Horizontal innovation, however, occurs when someone recognizes that two or more existing disciplines, techniques, or ideas can be combined in a way that was not previously appreciated. In horizontal innovation, the novelty is not in inventing entirely new methods, but rather in realizing that existing methods are compatible and can work together to produce better results. Regardless, both vertical and horizontal innovation result in a new method — that is, a new algorithm — that can solve a problem more effectively. We can refer to this as algorithmic novelty.

Where I disagree…

My disagreement with the article is with the suggestion that modern AI performs horizontal innovation and therefore exhibits algorithmic novelty. To understand why, we must make an important distinction between algorithmic novelty and output novelty. Output novelty occurs when an existing algorithm produces a result that humans had not previously encountered. Although output novelty can produce surprising and valuable results, it is not itself an act of innovation. Rather, it is a byproduct of the innovation that created the algorithm capable of producing those outputs.

Current AI systems excel at producing output novelty, but I do not believe they demonstrate algorithmic novelty. They do not independently invent new computational principles or discover fundamentally new ways of combining methods. Instead, they operate within a computational framework that has already been designed by humans. The model architecture, the learning algorithm, the optimization objective, and the training process define not only the space of possible outputs, but also the computational mechanisms the system can realize. The system is exceptionally good at searching this space, but the space itself is human-designed.

Chess engines

Consider the example of a chess engine. Modern chess engines frequently discover moves and strategies that surprise even the strongest human grandmasters. These discoveries are often described as “creative,” yet the engine has not invented the algorithm that makes these discoveries possible.

The algorithm behind the engine is itself an example of horizontal innovation. Researchers realized that tree search and pruning techniques could be combined to efficiently navigate the enormous search space of chess. They recognized that these two techniques complemented one another, producing a far more powerful algorithm. That conceptual leap — the realization that two methods could be combined into a superior computational strategy — was made by humans, not by the chess engine.

Once this algorithm existed, the chess engine simply executed it at a scale and speed beyond human capability. The surprising moves it produces are therefore examples of output novelty: they are new outputs generated by exploring a search space using a human-designed algorithm. They are not examples of the engine inventing a new computational method.

Evolutionary algorithms

Evolutionary algorithms provide another useful comparison. These algorithms often produce unexpected solutions, such as unconventional resistor or capacitor values when optimizing electronic circuits. However, the algorithms themselves are not innovating in the sense of creating new optimization principles. They repeatedly generate candidate solutions, evaluate them according to a predefined objective, and retain the best-performing candidates. Random initialization and mutation simply allow the algorithm to explore a vast landscape of possibilities. Again, although novel outputs can be produced, there is no algorithmic novelty (invention of a new algorithm), and thus no real innovation (horizontal or vertical).

Large language models

I would argue that modern AI operates in much the same way. Large language models (LLMs) and other deep learning systems search an enormous space of representations and combinations using optimization procedures that humans designed. The impressive outputs they generate demonstrate the effectiveness of these algorithms, but they do not demonstrate that the AI has invented new computational methods through either vertical or horizontal innovation.

This distinction is important because output novelty should not be confused with innovation. A search algorithm can generate outputs that no human has previously encountered, and those outputs may be surprising, useful, or even scientifically valuable. However, producing novel outputs does not imply inventing a new computational method. The novelty arises from exploring a search space using algorithms that were themselves created by humans.

Many of history’s greatest scientific and engineering breakthroughs were examples of algorithmic novelty. The invention of the Fast Fourier Transform, backpropagation, alpha-beta pruning, or public-key cryptography did not simply produce one more novel output within an existing framework; they introduced fundamentally new computational methods that changed how entire classes of problems could be solved.

Current AI systems have unquestionably demonstrated an extraordinary capacity for output novelty. What they have not yet demonstrated is the ability to autonomously create new computational frameworks or invent fundamentally new algorithms. Until they do, I believe it is more accurate to describe their achievements as remarkable search and optimization within human-designed frameworks than to describe them as genuine horizontal or vertical innovation.

Dr. Georgios Mappouras studied Electrical and Computer Engineering (ECE) at the National Technical University of Athens (NTUA), Greece. After graduating in 2014, he moved to the United States to pursue a Ph.D. in Computer Architecture at Duke University, which he completed in 2020. Since then, he has worked in the technology industry in Silicon Valley.