City Tutoring Discussion Forum

THE ONLY PHYSICALLY POSSIBLE BIOLOGICALLY REAL HUMAN RACES ARE DISCRETE HUMAN SUBSPECIES

Part One: The Fundamental Theorem of Taxonomy

It might interest you to know that of all, there cannot be such things as biologically real and biologically objective human races other than discrete subspecies which are biologically, zoologically, and ontologically discontinuous in every respect in which they are biologically, zoologically, and ontologically different from each other. I have discovered a truly remarkable proof of this theorem.

Note: By discontinuous I mean when a given multitude of things in such a relation to each other that on any side on which any two of the things in that given multitude differ from each other, there cannot be placed between these two any such third differing thing in that given multitude that the third such thing differs from each of the first two individually less than the first two differ from each other.

§1. Preliminary Principles of Scientific Classification

Before stating the Fundamental Theorem, it is essential to establish the foundational principles of a scientifically valid classification system:

Consistency and Uniqueness: The principles used to classify must be applied uniformly across all members of a genus.
Mutually Exclusive and Collectively Exhaustive Categories:
No member may belong simultaneously to more than one category at the same taxonomic level, and all members must be accounted for.
Use of Essential Attributes: Only traits fundamental to the essence of the living thing are valid criteria for classification.
Capacity to Absorb All Cases: The classification system must be able to include all known and yet-to-be-discovered members of the genus. These principles ensure that classifications are scientifically valid, unambiguous, and universally applicable.

§2. The Fundamental Theorem of Taxonomy

Theorem (Fundamental Theorem of Taxonomy): At any one and the same taxonomic rank, living things belong to biologically discrete natural kinds (taxa), which are discontinuous from each other in every respect in which their essential identities are essentially, intrinsically, biologically, and objectively different.

Proof:

Finiteness of Natural Kinds:

The number of distinct genomes—and thus distinct essential trait profiles (TDT profiles)—is finite and bounded, because DNA sequences have a finite number of base pairs (bounded above by Q). Or else, suppose there had been no upper bound on the number of base-pairs. But then there would be a DNA sequence with more base pairs than there are of estimated particles in the Milky Way Galaxy, which is clearly impossible! Therefore, there is an upper bound: call Q the least upper bound. Then the maximum number of possible ways to get a genome is 4 + 4^2 + ... + 4^Q = (4/3) · (4^Q ‒ 1), which is finite! Therefore, the number of natural kinds at any taxonomic rank is finite. Finitely many taxa cannot be infinitely divisible. Only infinitely divisible entities can form a continuum.

(Note: But infinite divisibility alone is not per se sufficient for a continuum. To understand the true nature of the continuum, the following preliminary definition is helpful: An admissible point or admissible indivisible individual of or within a given divisible person, thing, or being is an indivisible individual situated therein so that some path within the extent or among the appurtenances of the given person, thing, or being can pass or even retrace itself without interruption among all the indivisible individuals that conceivably exist at all. Any given divisible person, thing, or being is continuous if and only if:

* every piece, however small, can itself be broken into pieces each as small on all sides as one wishes (and this property we call Potentially Infinite Divisibility),
* every indivisible individual admissible within the given thing, region, or person is surrounded on all sides by the respective boundary of some piece, which is a piece of the given thing, region, or person, but which is as small as one wishes on all sides (and this property we call Fineness),
* the entire whole of the insides of the region cannot be broken up into pieces outside each other unless every such piece touches at least one other such piece (and this property we call Internal Contiguity),
* nor can any two such nonoverlapping pieces touch each other unless at least one of the indivisible individuals at which these two pieces contact each other and which is admissible within the given thing, region, or person, is also admissible within at least one of those two pieces (and this property we call Connectitude).

These 4 characteristics are each necessary and together sufficient and actually efficient to distinguish the continuous from the discontinuous.)

Taxonomical Determining Traits (TDTs) are hereditary aspects essential for defining taxa. A living thing’s TDT profile is the sum total of all TDTs, necessary and sufficient to determine its taxon.

Correspondence of TDT Profiles and Taxa: Each taxon at a given rank corresponds to a unique TDT profile; identical taxa share identical TDT profiles, and identical TDT profiles determine identical taxa.

Discontinuity of Essential Traits: If two taxa differed in an essential trait continuously, there would exist infinitely many intermediate TDT profiles. This is impossible because the total number of TDT profiles is finite. Hence, genetic variations defining different taxa must be discontinuous in every essential respect.

Mutual Exclusivity and Collectiveness: By definition of TDT profiles and their correspondence with taxa, each living thing belongs to exactly one taxon at a given rank, and all members are accounted for. Therefore, taxa at any given level of classification are finite, discrete, mutually exclusive, collectively exhaustive, and biologically discontinuous in all essential traits. QED

Correct me if I'm wrong here, but if I were to apply these definitions, then I wouldn't find any human subspecies. People move around and interbreed. For example, the Americas is full of Mestizos that are a mix of AmerIndian and Caucasian, and thus you will find an overlap between AmerIndians and Caucasians... but this ignores thousands of years where these two populations were geographically isolated. Furthermore, there may never have been a time period where any two races didn't interact and interbreed.

Now, I have seen PCA plots of ethnic groups, where the clustering closely matches the early 20th century's racial classifications. I do think researchers were onto something when they came up with 5 or 7 distinct human races.

You may disagree, and that's fine. Perhaps race is not the same as subspecies. My question to you would be what is race then?

mwtichen wrote: Mon Jul 06, 2026 9:48 am Correct me if I'm wrong here, but if I were to apply these definitions, then I wouldn't find any human subspecies. People move around and interbreed. For example, the Americas is full of Mestizos that are a mix of AmerIndian and Caucasian, and thus you will find an overlap between AmerIndians and Caucasians... but this ignores thousands of years where these two populations were geographically isolated. Furthermore, there may never have been a time period where any two races didn't interact and interbreed.

Now, I have seen PCA plots of ethnic groups, where the clustering closely matches the early 20th century's racial classifications. I do think researchers were onto something when they came up with 5 or 7 distinct human races.

You may disagree, and that's fine. Perhaps race is not the same as subspecies. My question to you would be what is race then?

The above objection made by mwtichen highlights a common misunderstanding regarding the intersection of taxonomic principles, historical geography, and modern statistical data. Below is the direct rebuttal of the points made in the objection. Furthermore, it relies on a spatial and statistical view of biology that misses the fundamental combinatoric reality of genetics and conflates phenotypic blending in a geographic space with the underlying ontological framework of taxonomy.

1. The Fallacy of Historical Isolation

* Isolation does not equal discontinuity.
* Geographic separation for thousands of years allows populations to drift.
* Drift creates statistical variations in allele frequencies.
* It does not create discrete, discontinuous biological boundaries.
* If a Mestizo population exists, continuity is actively demonstrated.
* The ability to blend seamlessly proves the lack of objective boundaries.
* Transitional populations cannot exist in a truly discontinuous system.

2. The Illusion of PCA Clusters

* PCA plots do not prove discrete subspecies.
* PCA spaces are highly sensitive to sampling methods.
* Sampling distinct geographic extremes artificializes the data.
* It creates visual gaps where actual geographic clines exist.
* Continuous clines look like distinct clusters when intermediate populations are left out.
* Statistical clustering is a mathematical tool, not an ontological reality.

3. Defining "Race" Under This Framework

* If race is not a discrete subspecies, it is not a biological reality.
* Instead, popular "races" are social and political categories.
* They use arbitrary thresholds on top of continuous geographic clines.
* A scientific classification system cannot rely on arbitrary thresholds.
* Therefore, continuous human variation fails the definition of objective race.

4. The Fallacy of Continuous Blending

* The forum member mwtichen cites Mestizos as proof of continuous biological overlap.
* This mistakes a mixture of discrete parts for a geometric continuum.
* A true continuum requires infinite divisibility, fineness, contiguity, and connectitude.
* Genetics is built on DNA base pairs bounded by a physical upper limit ($Q$).
* The maximum possible genome combinations form a strictly finite set: $\frac{4}{3}(4^Q ‒ 1)$.
* Finitely many states cannot form a mathematical continuum.
* Therefore, intermediate populations are collections of discrete, discrete genetic points, not a continuous blend.

5. Why PCA Clusters Are Ontologically Irrelevant

* The proposed objection made by mwtichen points to PCA plots matching 20th-century racial categories.
* PCA plots project high-dimensional data onto continuous geometric coordinates.
* This smooth mathematical projection is an artifact of the statistical tool.
* It does not alter the discrete, finite nature of the underlying Taxonomical Determining Traits (TDTs).
* Statistical proximity on a chart does not equal ontological continuity in nature.

6. Resolving the Question: "What is Race Then?"

* The proposed objection made by mwtichen poses the question of what the term "race" would mean if it didn't mean a continuous subspecies.
* Under the Fundamental Theory of Taxonomy, each and every individual valid biological taxon respectively requires some suitable unique, discrete TDT profile.
* If "races" interbreed to create a finite set of intermediate discrete genomes, they fail the test of essential discontinuity at that rank.
* Even worse, a continuum of such differing classes as at any taxonomical level would require an actually infinite number of such differing classes, and therefore more unique TDT profiles than the principles of genetics at the molecular level will ever allow given that there is an upper bound on the number of such combinatorically possible genomes.
* Therefore, the arbitrary boundaries drawn around continuous clines are social conventions.
* They are not biologically discrete natural kinds.

If any of my fellow forum members finds any step of my proof fallacious, well then I invite him or her to discover the fallacy.

Casus_Conscientiae wrote: Tue Jul 28, 2026 1:05 pm The proposed objection made by mwtichen poses the cavillous question of what race is if it is not a continuous subspecies.

I'm going to tell you this only once. Don't presume to understand my motives when responding to your post and don't mischaracterize my response. I will not tolerate disrespect.

mwtichen wrote: Sat Aug 01, 2026 2:00 pm

Casus_Conscientiae wrote: Tue Jul 28, 2026 1:05 pm The proposed objection made by mwtichen poses the cavillous question of what race is if it is not a continuous subspecies.

I'm going to tell you this only once. Don't presume to understand my motives when responding to your post and don't mischaracterize my response. I will not tolerate disrespect.

That is why I have recently today deleted the word "cavilous" and wrote instead: "The proposed objection made by mwtichen poses the question of what the term "race" would mean if it didn't mean a continuous subspecies"

Casus_Conscientiae wrote: Mon Aug 03, 2026 11:50 am That is why I have recently today deleted the word "cavilous" and wrote instead: "The proposed objection made by mwtichen poses the question of what the term "race" would mean if it didn't mean a continuous subspecies"

Thanks