*Later edit* (17th of March 2021):

The distinction between *invention* and *discovery* in this context is a confusion induced by the assumptions of materialistic determinism.

Invention is the organized act of pursuing discovery through exploration. Discovery is just the achievement (and the popularization) of such pursuit, when it is successful.

To this I would add that the frontier of mathematical discovery is strongly-correlated with our (collective, as a civilization) ability to sift through *pathological mathematical cases*** for *patterns deemed useful for the observing-thinkers and their kin/kind*

Remember that is the 14th century there were no imaginary numbers. Neither was there studied within mathematics any notion of non-Euclidean geometry, curvature or curved space.

Mathematical discovery is just the *observation, validation and publication* of ways to formally conceptualize that which had yet to be conceptualized previously by others.

In mathematics, a pathological object is one which possesses deviant, irregular or counterintuitive property, in such a way that distinguishes it from what is conceived as a typical object in the same category. The opposite of pathological is well-behaved.

https://www.wikiwand.com/en/Pathological_(mathematics)

*Original article:*

My pre-existent bias (the one we oftentimes refer to as “instinct”) is that mathematics is being discovered, as it would make it more pure, a thought which is ennobling, hence sought-after and mentally-seducing. However, upon further inspection, I start to realize the many anthropocentric biases that limited or directed or dictated the building of axiomatic systems throughout time. Should one accept the define mathematics as the outrospective, introspective and analytical means to create algorithm-like (algoritmoids) mental tools that allow us to make extremely-high confidence inductory and deductory predictions. The essential difference that makes these tools “algoritmoids” is that they (by intentional lack of specification) lack one essential quality of algorithms: they do not deliver the a priori promise of halting (time bounded) and they do not deliver the a priori promise of sure or **almost-sure delivery **or **almost-sure correctness** .

Making this difference seems essential *(so as to avoid the Christianity-enforced bias of saying “crucial”) because algorithms need to guarantee that **they finish in some finite** (preferably know, preferably polynomial, preferably short or at least shorter than the human-observer lifetime) **time**.

Types of almost-sure deliveryomni-presence/always-availabilityp(result|call) == 1

orpseudo-omni-presence/always-availabilityp(result|call) ~= 1

Types of almost-sure corectness

Functional probabilistic idempotencep( call_i( {F}, {D} ) == call_j( {F}, {D}) ) ~> 1

Thus mathematics, we propose, is neither entirely being constructed not entirely being discovered. The axioms are being constructed by inference of likely relationships observed in the local universe, so they formulation and acceptance is contingent upon the coherence of the experience of the proposing sentients. The implications of the axioms are being discovered by repeated mental simulation, involving repeated attempts to postulate the outcome of future interactions and to validate such outcome against the a posteriori mental experiment/validation/rule enforcement.In order for a proof to be deemed valid (by the mathematical community) the path from hypothesis to conclusion must be described only in those steps which had been previously accepted* within enriched with its often-implicit axiomatic support.

One has to ask one what grounds (implicit, factually-repetitive or intuitive) does one accept an axiom as valid, relevant, reasonable?Does not such axiom, in order to be deemed wholly, have to be recognizable, as pattern or as collection of relationships, from the local-universe oft observed of the judging observer(s)?The observer-creator is only involved by that in which our conscience, which analyzes the implications and assertions of mathematics, has emerged from the universe that said creator allegedly bootstrapped/created/defined/is running.

Mathematics seems to be discovered, nonetheless, the same way the perfect form for a a cutting tool was discovered by the primitive man. Just as the tool, mathematics still represents a concoction of essential secrets of the Universe and circumstantial limitation of the human body, mind and perception.Things that seem to be arbitrary in math:

- Space is
**straight**(the fourth postulate of Euclidean geometry) **Naming concepts**and**variables**– Symbolism, phoneme etymology and lexome etymology- Influenced by availability and frequency within the vicinity of the user

**Linear-centric writing**(probably based on information capacity and saliency constraints of the foveal region in the mammalian eye)- Left-to-right writing (cerebral laterality )
- Up-to-down writing (form on hand/joints; gravitational bias)

**Decimal base**(based on number of fingers)- The numeral base may be relevant as it would seem to indicate a junction/link/mapping between the abstract mind and homoluncus, which contains ten distinct spatial items, one for each finger. This is why most children instinctively use their fingers as a “external, partially-stable state” when learning to count.
- In the beginning of autopoietic, ahead-of-time simulated type of intelligence, maximizing the numeral basis also meant maximizing the capacity to store

**Arithmetic**- Sequence of operations (convention on the ordinal relationship between operators in linear-centric writing)
- Heuristic: operations which are most often used become less salient (less special, such as addition and subtraction), so are left for last; operations which were later defined (i.e. exponents) are deemed more salient, so are performed first;
- it would seem that mathematical functions have

- Arrow-headed vectors to indicate direction
**Idempotence**of values and relationships (they stay the same upon repeated inspections)**Set theory**: Boolean/binary/quantized /discrete-valued,scalar,time-invariant, zero-dimensional . Alternative: Fuzzy set theory and subjective set theory.**Boolean Logic**: /binary/quantized /discrete-valued,scalar,time-invariant, zero-dimensional logic. Alternatives: Multi-value logic, Subjective logic, fuzzy logic;)- Functional association is unique
- Choosing the criteria by which
**elementary function**(aritmetic, sin, cos, exp)- https://en.wikipedia.org/wiki/Nonelementary_integral
- https://en.wikipedia.org/wiki/Hypertranscendental_function -> in this case, the label “hypertranscendental” / “transcendentally transcendental” is relative to the power of analysis, inference and resolution that existing mathematical utilitary insight poses.
- https://en.wikipedia.org/wiki/Hypertranscendental_function

**Zermelo-Frankel**:**Axiom of choice**– we once again choose to believe if we cannot yet imagine something then it must not exist;

Things that would be dificult to establish in the infrastructure of communication with a different species:

- The Spectral Encoding: Electromagnetic Spectrum band set —to—> symbol-space mapping
- Frequency Encoding
- Amplitude Encoding
- Pulse Encoding
- #choose the mechanism which is computed to be most robust for interstellar communication

- The Temporal Encoding (see frequency band)