The Orchid Project aims to replace a wide swath of modern mathematics with a set of digital structures that can be understood and manipulated by humans and computers alike. To understand how Orchid will fulfil this goal, it is fruitful to first consider the nature of mathematics. Mathematics is the study of abstract entities with stable characteristics and behaviors. Over the past millennia, human beings have developed a set of written symbols used to describe the characteristics and behaviors of these abstract entities. These written symbols allow humans to reference properties of the abstract entities being studied and perform symbolic manipulations on these entities in accordance with well-defined rules.
Modern mathematics is a symbiosis between the human mind and the aforementioned symbol set. Without the human mind, the symbols are essentially worthless and only interact with reality in accordance with physics of their constituent physical materials. Without the symbols, humans would have to reason about abstract entities without any outside assistance, and therefore would suffer from the limitations of human memory and intelligence. Together, however, there is a beautiful symbiosis. The abstract structures live within the human mind, but they can be compressed and stored compactly within the symbol set of mathematics.
However, this brings us to an interesting philosophical question: why is mathematics useful? Mathematics is useful simply because there is stable structure in reality as perceived by humans. The term "stable" is defined here to simply mean "existing for a non-trivial duration of time". While there are, of course, no guarantees about the stability of the different entities perceived within reality, there nonetheless seem to exist a very large number of entities that exhibit some degree of stability.
One particularly fundamental reason contributing to humanity's evolutionary fitness is the human mind's ability to create a model of the stable structures in the human’s environment, and act according to this model. If, for instance, a human learns there is a meteor directly overhead, the human will use their internal model of the world to reason that they must run away in order to survive.
While the importance of the brain's ability to harbor an internal representation of reality cannot be understated, humanity has progressed even further by creating spoken language. Spoken languages allow groups of humans to translate the representations stored within their brains into a set of stable auditory signals. These auditory signals are then decoded by other humans and translated into neural representations of the world.
Forming and reasoning about internal neural representations of the world takes time and effort, and frequently humans form the same sorts of representations. For example, even without communicating, multiple humans could easily learn that a meteor overhead probably means grave danger. Spoken language is an incredibly tool because it decentralizes the effort required to formulate internal representations of reality. As a simple example, a person entering India for the first time several millennia ago might never have interacted with a tiger before. The person could either learn about the dangers of tigers by experiencing one for himself/herself, or he/she could communicate with the locals and learn that the big striped orange and white cats ought to be avoided. The latter option obviously better lends itself to human survival.
Spoken language is a miraculously useful tool, but it suffers from the fact that audio signals decay rapidly, thus requiring two humans to be in immediate contact while communicating. This limitation is addressed by written language, which is next in the chain of incalculably useful human developments. The written word can last for centuries or even millennia, and therefore allows humans beings to share neural representations of reality across wide swaths of time.
So where does mathematics fit into all of this? Languages like English or Mandarin allow humans to describe a large portion of either their perceived reality or even hypothetical situations, and typically rely on some context for understanding. “How fast did Blake run?” I might ask. “Fast,” or “slow,” or “faster than Usain Bolt,” you might answer. Given my current context, I’ll probably form an internal representation of the events you witnessed that isn’t too far off from what you actually perceived. However, the words “fast,” “slow,” and even “faster than Usain Bolt” are all very imprecise, and effectively lose all meaning without the ill-defined context I described.
Mathematics attempts to address this lack of precision by describing both the structures and context in terms of abstract imagined entities with infinitely precise characteristics and infinite stability. To my question of how fast Blake ran, you could instead answer “44.83 ± 0.9 km/h.” You could also provide me with a mathematical model of Blake’s trajectory using polynomials and describe the Blake’s physics using Newtonian mechanics. Given the definition of kilometers, hours, and real numbers, someone 300 years from now would still be able to form a highly accurate internal representation of Blake’s speed, were they to desire that knowledge.
Mathematics therefore gives humans the ability to describe reality with far greater detail, precision, and accuracy than languages like English or Mandarin. The toolset of mathematics has also informed the development of some (if not all) of humanity’s most impressive modern technologies.
Why, then, has mathematics not replaced the languages of the world? If mathematics can provide such superior descriptions of reality, why don’t we entirely replace “the old technologies,” like English? The reason we haven’t done this is because mathematics comes with a high cost. It’s really, really hard. The reason why “Einstein” has become synonymous with “genius” is because Einstein formulated a mathematical description of the world that was more consistent with reality than previous attempts. Mathematics isn’t nearly as forgiving as world languages. While humans can rely on mutual context to convey information with language that would otherwise be imprecise or inaccurate (think metaphors or sarcasm), well-defined mathematics doesn’t allow for any of that behavior.
Before I continue, I’d like to once again emphasize how critical both language and mathematics have been to the improvement of the human condition. Even though I will talk about how Orchid aims to achieve a superior technology, the importance of both mathematics and language should never be taken for granted.
So where does all this leave us? Spoken/written languages can describe a broad swath of reality, but they are imprecise and typically rely on ill-defined and brittle context to actually convey meaning. Mathematics can describe some aspects of reality in incredible detail, but it is difficult to use and struggles at generalizing to complicated systems.
The Orchid Project aims to move beyond language and mathematics for formulating representations of reality by utilizing advances in computers to significantly lower the costs associated with mathematical descriptions of reality. As previously described, right now mathematics is a symbiosis between the human mind and a symbol set, wherein the actual mathematical structures live in neural representations within the human mind but can be represented compactly within a set of written symbols.
Put in high level terms, Orchid transforms neural representations of mathematical entities into computer data structures, which can be created and manipulated by computers. While this is easily said, the ramifications of this statement are enormous. By making this translation from the brain to the computer, Orchid allows computers to “think” about mathematics like humans do.
While you could probably see the importance of this concept after a moment or two of thought, let me perhaps provide some motivation for why this could be revolutionary. Modern computer architectures are capable of roughly a billion operations per second. The spiking rate of a neuron in the human brain is about 200 spikes/second. If we equate a single spike as being approximately equal to a single clock cycle, then a computer processes information more than 5 orders of magnitude faster than the human brain.
What this means, in more accessible terms, is that a task that would take the human mind 1,000 years could be completed by a modern computer in under 4 days. This example obviously makes some irresponsible assumptions about the similarities between the human brain and computers, but the point being made remains valid.
With that said, by drastically reducing the cost associated with mathematical descriptions of reality, Orchid aims to give human beings a tool that robustly addresses the limitations of both languages and mathematics in formulating representations of reality. In accordance with the historical trend outlined throughout this text, one can hope that the tools provided by Orchid can aid in both a drastic improvement of the human condition and the progression of organized complexity within reality as a whole.