The sphere-and-plane argument; point versus part; the mustard-seed comparison; separability, infinite divisibility, al-Razi’s suspension of judgment, and geometric objections Read the bilingual lesson for The Indivisible Atom: Arguments, Infinite Division and Objections and test your understanding with a focused quiz.
Lesson
1. The precise problem of an indivisible part
Al-jawhar al-fard is posited as a substance admitting no further division—actual, imagined, or stipulated. The question is whether a body is ultimately constituted by such indivisible units or whether magnitude remains divisible without a final unit.
A familiar kalām position: division of bodily parts reaches an indivisible limit.
The philosophical position under discussion: a magnitude can remain divisible, so a body need not be an aggregate of actually indivisible atoms.
An object being the smallest we can observe is not the same proposition as its being logically or metaphysically indivisible.
2. Competing accounts of body
وعند الفلاسفة لا وجود للجوهر الفرد
According to the philosophers under discussion, an indivisible atom does not exist.
One account explains a body through ultimately indivisible substances. The competing classical account uses prime matter and form and treats extended magnitude as still divisible.
The word part must not be silently shifted between these accounts. In one it names an actually existing ultimate constituent; in another it can refer to a division that may be performed or conceived. An argument must state which meaning it needs.
3. A true sphere placed on a true plane
لو وضعت كرة حقيقية على سطح حقيقي
Suppose a perfectly true sphere is placed on a perfectly true plane.
A celebrated geometric demonstration argues that an ideal sphere touching an ideal plane must meet it at an ultimate, non-extended contact. If the contact covered an extended line or area, the perfect sphericity assumed by the proof would supposedly be compromised.
The intended conclusion is not merely that a geometric contact point exists, but that it supports an actually indivisible bodily constituent. The question is whether the proposed bridge from a geometric point to a bodily atom is valid.
4. What follows from the number of contacts?
لم تماسه إلا بجزء غير منقسم
It would touch it only by an indivisible part.
The proof assumes that contact across two or more constituents would create an actual extended line or surface at the contact and conflict with ideal roundness. The contact is therefore treated as indivisible.
Yet two questions differ: Is contact represented geometrically by a point? and Is that point an independently subsisting physical atom? The first does not entail the second without a further premise.
5. Point, line and substance are distinct
A geometric point is treated as a location without extension. A line has one-dimensional extension. A bodily substance, however, is an additional claim about external constitution and subsistence.
Even if the sphere argument gives an indivisible point of contact, one must still show that this point exists as a bodily atom rather than as the boundary of contact.
Example: conceiving the endpoint of a drawn line does not establish an independent physical particle at that endpoint. This is the objection's logical core.
6. An actual line versus a possible line
An actual line is taken as an existent, determinate extension. A potential or postulated line can be marked out within a geometrical analysis without requiring an already separate material component.
This distinction is used against the sphere proof: a geometric boundary of contact does not automatically amount to a collection of independently existing actual substances.
Potential divisibility, an accomplished division, and already constituted independent parts are three separate propositions.
7. Why the sphere proof is not decisive
فإنه إنما يدل على ثبوت النقطة، وهو لا يستلزم ثبوت الجزء
It establishes at most a point, and the existence of a point does not entail the existence of an atom.
The objection identifies a shift from the kind of entity established by geometry to the kind asserted in metaphysics. A geometric point is not interchangeable with a physical or kalām substance.
The atomist still needs a further premise: every true contact must be realized through an independently subsisting indivisible material unit. The definition of an ideal sphere alone does not establish that premise.
8. Two prominent demonstrations reported in kalām
وأشهرها عند المشايخ وجهان
Two arguments are especially prominent among the kalām scholars discussed here.
The first compares objects of very different magnitudes, illustrated by a mustard seed and a mountain. The second examines whether bodily parts can be joined or separated.
The first concerns magnitude and an allegedly infinite number of parts; the second relies on divine power, possibility and separation. Their premises and replies must be examined separately.
9. The mustard seed and the mountain
لم تكن الخردلة أصغر من الجبل
A mustard seed would not be smaller than a mountain.
The argument says that if every body were an actually assembled infinity of further-divisible parts, both a seed and a mountain would contain infinitely many. How then could one be genuinely less in size?
Two premises require scrutiny: 1. unlimited divisibility entails an already-existing infinity of constituents; 2. any two purported infinite collections must be equal in measurable size. Both are contested, so the example is not itself the conclusion.
10. Divisibility does not imply an actual infinity of atoms
Unbounded divisibility can mean that wherever one marks a boundary in a magnitude, a further subdivision remains possible. It does not by itself mean the magnitude has already been assembled from infinitely many independent substances.
Example: a line may be halved repeatedly; performing or imagining further halvings is not the same as discovering a pre-existing physical particle at every potential boundary.
This is a central response: indefinite potential division is not interchangeable with an actually constituted infinite aggregate of parts.
11. Joining and separation: the second proof
The second proof begins with bodily parts being united or separable. If their union is not essential in such a way as to prevent all separation, their separation is described as possible and therefore within divine power.
If each separated part could be split again, the proof seeks a final indivisible terminus and calls it an atom.
The real dispute is not over divine power as such. It is over whether the successive acts of separation posited by the proof are intelligible possibilities and whether their possibility entails a terminal part.
12. Divine power and the domain of possibility
In the theological discussion of divine power, the object of power is what is possible. A self-contradiction is not simply a difficult action requiring more force.
The proponent argues that if separation is possible, divine power extends to it. The critic asks which separation is meant: physically splitting already existing constituent parts, or assigning further boundaries within a continuous magnitude?
Until the object of the proposed act is specified, the principle of power does not itself establish an indivisible bodily atom.
13. Does possible separation establish a final atom?
The proposed chain is: union is not inseparable from the parts; separation is possible; possible separation is within divine power; therefore separation can reach a last indivisible unit.
The objection targets the final step. The possibility of some division does not prove that every conceivable division has already occurred or that a sequence of division must terminate in a particular material atom.
Example: the capacity to halve any specified segment further does not itself imply the existence of a final unhalvable segment. That would require an additional argument.
14. Points and the problem of inherence
Once a point is admitted as a boundary or a location within a line or body, one must ask what kind of dependence it has on that host. If it is merely a limit or endpoint, it has not been shown to be a separately subsisting bodily atom.
The discussion distinguishes ways of speaking about something being “in” another. Such a relationship need not mean that both are constituent physical particles of the same kind.
Thus the existence of a limit cannot simply be converted into proof of an independent indivisible substance, including in the sphere argument.
15. The philosophical reply: no preassembled atomic aggregate
بل يقولون: إنه قابل لانقسامات غير متناهية
Rather, they say it admits indefinitely many divisions.
The key is capacity. The body is not being described as an already assembled infinity of physical particles, but as an extended magnitude admitting further potential subdivision.
To attack that account as though it affirmed an already existing infinite inventory of atoms would alter its premise. The distinction between a possible boundary within a continuum and a physically separated constituent is decisive.
16. Actual, imagined and postulated division
Actual division separates a thing into parts. Imagined division considers its parts mentally, while postulated division assigns boundaries as a supposition without performing a cut.
An indivisible-atom thesis may deny further division in all these senses. A continuity thesis can allow further conceived or potential subdivision without affirming already-existing discrete atoms.
Consequently, “Can this be physically split?” and “Can two regions still be conceptually distinguished?” are not identical questions.
17. The arguments against atoms also face criticism
وأما أدلة النفي أيضا فلا تخلو عن ضعف
The arguments denying an indivisible atom are also not free of weakness.
The evaluation runs in both directions. Just as a geometric illustration for atoms may not prove them conclusively, arguments denying atoms from continuity, points or indefinite division also require careful premises.
Showing that an affirmative proof fails is not the same as establishing the negation. Each side bears responsibility for the inference it makes.
18. Challenges to unending subdivision
Kalām objections to endless divisibility invoke imagined cases involving ever smaller bodies or finer lines, sometimes appealing to touch, motion and the order of parts.
The philosophical reply distinguishes practical ability to cut a material body from the intelligibility of further division within its magnitude. A physical limitation on cutting is not automatically a demonstration of a metaphysical last unit.
The analysis must not confuse practical impediment with logical impossibility, or logical possibility with an accomplished physical event.
19. A geometrical circle drawn by rotation
An argument in classical geometry constructs a circle by keeping one endpoint of a line fixed while rotating the other. The moving end traces the circumference, and the fixed end is treated as the center.
This construction explains a geometric relation, but the metaphysical question remains: does that construction and the continuity of the motion prove the existence of indivisible material substances in the body?
Critical distinction: a center, circumference and rotating radius can be defined within a geometrical model. Inferring the presence or absence of ultimate physical atoms requires an additional and justified argument.
20. Al-Rāzī’s suspension of judgment
مال الإمام الرازي رحمه الله في هذه المسألة إلى التوقف
Imam al-Rāzī inclined toward withholding a definitive verdict on this issue.
Tawaqquf means declining to affirm or deny conclusively when the available demonstrations do not reach certainty. It is an assessment of the evidence, not simply abandoning thought.
Two lessons follow: 1. a persuasive argument is not necessarily a demonstrative proof; 2. withholding judgment on this specialized question does not suspend all other theological claims.
21. Prime matter and the argument for creation
Affirming an indivisible atom, rejecting the classical account of matter and form, and demonstrating the world's origination are related but distinct conclusions.
A criticism of one theory of bodily constitution does not instantly refute every alternative. Likewise, a conclusion about divisibility does not by itself establish temporal beginning without further premises.
The methodological question is how a claim about internal bodily constitution is connected by argument to a judgment about origination or eternity.
22. Arguments based on classical geometry
وكثير من أصول الهندسة
And many foundations of geometry.
The discussion invokes circles, spheres, points, surfaces, and motion. Some classical demonstrations suppose that unlimited divisibility threatens particular geometric constructions or an account of uninterrupted motion.
One must distinguish a geometric idealization, an assertion about external bodies, and a theological conclusion. A geometric model establishes a claim about physical constitution only if the bridge between them is justified.
23. Classical celestial-motion arguments
دوام حركات السماوات
The supposed perpetual motions of the heavens.
Classical philosophical arguments sometimes assumed a particular system of celestial spheres and uninterrupted circular movement, then drew consequences for geometry, motion and bodily constituents.
These assumptions are historical cosmological premises, not established findings of contemporary astronomy. The strength of any inference from them depends on those premises, and their use should not be confused with direct observational proof.
24. Rupture and rejoining in heavenly bodies
وامتناع الخرق والالتئام عليها
And the alleged impossibility of rupture and rejoining in them.
In a classical celestial-sphere model, kharq means tearing or rupture and iltiʾām the reunion of separated parts. The model sometimes declared these changes impossible in heavenly spheres and drew conclusions about their constitution.
The response is to demand grounds for that special claim. An impossibility stipulated within one ancient astronomical framework does not, without further proof, yield a universal claim about all physical bodies.
25. Assessing the arguments for and against atoms
The indivisible-atom debate has brought forward the sphere-and-plane case, the seed-and-mountain comparison, union and separation, and geometric and celestial-motion arguments. Each depends on distinct premises.
Central distinctions are point versus material constituent, actual parts versus possible cuts, divine power versus the possibility of its object, and geometric model versus external substance.
These distinctions enable a reasoned evaluation of affirmation, denial or suspension of judgment. The contrast with accidents (aʿrāḍ) remains fundamental: substances and their constituents concern bodily constitution, whereas accidents are characterized by dependence on a substratum.
Practice questions
Question 1
Which definition matches al-jawhar al-fard in this discussion?
- A subsisting atom admitting no further division
- A divisible surface classified as an accident
- A magnitude that is only practically uncuttable
- A part unseparated in fact but divisible by supposition
Question 2
What conclusion does the ideal sphere–plane proof seek?
- That every contact contains an actual line
- That contact establishes an independently indivisible material part
- That every point is necessarily an accident in a substratum
- That a true sphere cannot have a circumference
Question 3
Which distinction underlies the main reply to the sphere proof?
- The point of contact versus the sphere’s diameter
- A real versus an idealized plane
- A point of contact versus a subsisting substance
- The contact boundary versus the radius
Question 4
What mistake follows from equating an actual line with a merely possible line?
- A stipulated division will be regarded as impossible
- Every line will be treated as a self-subsisting accident
- The distinction between potential and actual extension will be clarified
- A proposed boundary will be treated as an already existing material constituent
Question 5
What is the first contested premise in the mustard-seed–mountain argument?
- That potential unlimited divisibility entails actually infinite constituents
- That imagined boundaries and actual constituents are equally numerous
- That mere naming of parts fixes volume
- That substances always turn into accidents
Question 6
Which reply best addresses the seed–mountain comparison?
- Both sizes depend only on contact points
- Relative magnitude is not necessarily identical to the number of possible cuts
- Anything potentially divisible stops being a body
- Magnitude is intelligible only after enumerating separated particles
Question 7
Which step of the union–separation argument requires further proof?
- That every body has actual continuity
- Calling connection an accident
- That possible separation must end at an ultimate indivisible part
- Treating separation as a change of motion
Question 8
What must be established before applying the divine-power premise?
- Whether every separation occurs without motion
- Whether every proposed object can be considered outside power
- Whether extension and place must have one definition
- The meaning and coherent possibility of the proposed separation
Question 9
Which conclusion does repeated imagined division not directly establish?
- The prior existence of infinitely many independent physical atoms
- Positing a conceptual boundary
- Supposing half of an extended magnitude
- The possibility of another conceptual subdivision
Question 10
Which is an example of actual rather than merely conceived division?
- Marking the midpoint of a rectangular plane
- Physically cutting wood into two separated pieces
- Considering another division of an existing part
- Mentally determining two halves of a line
Question 11
What does the nonextension of a geometric point fail to establish?
- A nonextended geometric location
- Calling a boundary of a surface a point
- Its being an external independently subsisting bodily atom
- Using a point to describe sphere–plane contact
Question 12
What fallacy arises from confusing indefinite divisibility with actually infinite parts?
- Separating relations of magnitude and volume
- Distinguishing actual from stipulated division
- Separating boundaries from body components
- Objecting to potential subdivision as though it were an actual atomic assembly
Question 13
If a proposed positive proof fails, what necessarily follows?
- Only that this particular proof is not conclusive
- That the rival definition must be contradictory
- That the negation is decisively proved
- That all other arguments are automatically invalid
Question 14
What is the appropriate meaning of al-Rāzī’s tawaqquf here?
- Categorically affirming despite weak proof
- Withholding a categorical verdict because the arguments are not decisive
- Treating denial as necessary because affirmation is unproved
- Treating every technical problem as essential creed
Question 15
What is the fundamental objection to inferring atomic constituents from the rotated-radius construction of a circle?
- A center can never be designated
- Rotation cannot produce a geometric circle
- A geometrical circle does not itself entail independent indivisible physical atoms
- A rotating radius must remain entirely motionless
Question 16
How should claims about classical celestial spheres be handled?
- Treating all historical geometric premises as externally proven
- Excluding geometry and astronomy in every case
- Treating a cosmological model as direct proof of bodily constitution
- Treat them as historical premises and evaluate the conditions of the conclusions
Question 17
Which pair correctly explains kharq and iltiʾām?
- Rupture; rejoining of separated parts
- Joining; then separating
- Rotation; then rest
- Increase of parts; then decrease
Question 18
How is a conclusion about the world's origination related to bodily constitution?
- Equating prime matter with origination
- Further valid premises are needed to connect bodily structure to origination
- Assuming atomism immediately establishes origination
- Making volume alone the proof of origination
Question 19
Why are point and substance not interchangeable in the sphere objection?
- Because a point is always a self-subsisting accident
- Because substance and accident differ only by color
- Because a point is a limit while substance involves independent external subsistence
- Because geometric lines must subsist as accidents in bodies
Question 20
Which statement pinpoints the response to the mustard seed comparison?
- Every possible division leaves equal volumes
- All actual parts of a thing have equal size
- Geometry has no relevance to the seed–mountain example
- Different magnitudes and indefinite possible divisibility are different claims
Question 21
What is needed to move from a geometric model to a physical conclusion?
- A justified connection and supporting premises
- Applying the model to an external body without further evidence
- Drawing the model repeatedly
- Treating a postulated line as an actual substance
Question 22
Which pair belongs to the competing philosophical account?
- Substance and accident
- Prime matter and form
- Part and whole
- Accident and substratum
Question 23
What are the core notions of the second prominent kalām proof?
- Matter and form
- Point and sphere
- Union and separation
- Largeness and smallness
Question 24
What does a point being located in a host not immediately entail?
- A point being an endpoint of a line
- A point as a geometric boundary
- A point’s relation to a location
- That the point is an independently subsisting particle
Question 25
Which conclusion best reflects the closing evaluative stance?
- Evaluate both sides and avoid unwarranted certainty
- Immediately assume affirmation is true
- Immediately assume denial is true
- Ignore distinctions between the two sides’ premises
Question 26
If an imagined subdivision has not been shown to occur physically, what follows for the indivisible-atom question?
- The body must already contain infinitely many separated parts
- The relation between possible division and actual bodily constituents needs further argument
- It necessarily proves the classical celestial-sphere model
- It renders every physical division impossible
Question 27
Establishing a geometric point does not by itself prove an independently subsisting physical atom.
Question 28
Indefinite potential divisibility necessarily means a body is already composed of infinitely many discrete atoms.
Question 29
Al-Rāzī’s tawaqquf can express a scholarly judgment that neither side’s evidence is conclusive.
Question 30
The classical claim that celestial spheres cannot rupture or reunite is an established law of modern astronomy.
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