The hypotheses require (some degree of) faith.
Let us assume the starting point Descartes did when trying to show what even one skeptical of any faith or belief must by mere logic alone accept. That is, if we are free to doubt all sensory input and suppose we are under the spell of a magician or (in a more modern context) trapped in the Matrix such that we cannot assume anything we take to be real actually is, what (if anything) might we still be able to assert is necessarily true? Alternatively, is there anything which (even assuming such radical skepticism) that we can't deny?
Descartes and I agree that we must accept at the very least that we cannot logically answer "yes" to the question "is it possible that I myself do not exist?", because I, the one asking the question, must exist in order to answer the question.
With such an approach, faith is required to accept just about anything, and that was Descartes intent.
Numbers, however, present an interesting case. A great (if ignoble and immodest) mathematician by the name of Kronecker once said "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk" or "The whole [entirety of the] numbers has the loving God made, all else is the work of man" (my more literal translation; it is usually translated "God made the integers, all the rest is the work of man"). Stephen Hawking used part of this quote as the title of his book on the history of mathematics. It is, however, almost always misunderstood. Kronecker wasn't saying something profound about God or really about math. He was responding to another great mathematician (Cantor) and those like him. He believed that the only numbers which "exist" are the integers (denoted by the symbol
for the German "Zahlen"). He was a constructivist, meaning that he believed numbers like pi or anything that wasn't an integer or explicitly constructed out of an integer (e.g., 1/3 is a ratio of two integers). This point of view has almost no supporters today (I have met only one: N J Wildberger) because the advances in mathematics and their use within the sciences (particularly physics) has placed the real numbers and the real number line on a rigorous (logical) footing.
However, a bigger issue is what it means for a number to exist at all. In practice, it is perfectly reasonable to devise all kinds of mathematical entities, number systems, etc., such as Conway's surreals, the p-adic completion of the rationals, the ordinals, etc. And it is perfectly reasonable to say that none of them exist (I believe that most people who aren't philosophers of mathematics or mathematically inclined philosophers hold this position implicitly, while those who are or who are deal with mathematics as physicists like Tegmark and Penrose do implicitly or explicitly assume a Platonic-like reality for mathematical entities).
The issue here, however, need not concern ontology. I need not assert that 0.999... exists or that 1 exists to assert that the two are equal. That's what makes mathematics a special case when it comes to issues like faith in what exists. Just as logic still works even in the radical skepticism of Descartes thought experiment, so too it works if we assume no numbers actually exist. Whatever ontological status numbers may or may not have, they have a logical status.
You have accepted the number 1 it seems, and the number 0.999..., but because you do not take these to be equal you have been forced to posit the existence of a number with illogical properties: a number that is smaller than all other numbers, which would make it smaller than itself and/or a number that when divided by 10 yields the same number but isn't 0. We are here dealing only with logic, not existence.
I am using faith to mean: complete trust or confidence in someone or something (or what my computer's dictionary and I imagine most dictionaries say in conveying the definition of faith)
If your knowledge of philosophical and/or social science literature is sufficiently advanced such that you are familiar with terms like intersubjectivity, surely you are aware that dictionaries are at best guides to usage and do not define words (which are always and everywhere polysemous). The following is from the entry "faith" from the most authoritative and comprehensive dictionary of the English language (for brevity, I have omitted the examples of use from all but first sense, as in each case they date from the 1300s to the 21st century) :
"III. Belief, trust, confidence.
5. Belief in and acceptance of the doctrines of a religion, typically involving belief in a god or gods and in the authenticity of divine revelation. Also (Theol.): the capacity to spiritually apprehend divine truths, or realities beyond the limits of perception or of logical proof, viewed either as a faculty of the human soul, or as the result of divine illumination.
Earlier evidence refers almost exclusively to the Christian religion, divine revelation being viewed as contained either in Holy Scripture or in the teaching of the Church. In this context faith is often considered in relation to justification before God, and contrasted with works. Cf. justifying faith at justifying adj. Special uses.
▸c1384 Bible (Wycliffite, E.V.) (Royal) (1850) James ii. 17 Feith [L. fides], if it haue not werkes, is deed in it silf.
c1405 (▸c1380) Chaucer Second Nun's Tale (Hengwrt) (2003) l. 64 Feith is deed with outen werkis.
▸c1443 R. Pecock Reule of Crysten Religioun (1987) 425 Feith is a knowingal vertu—þat is to seie such where wiþ we knowen sum treuþe and is þe knowing of þe same trouþ.
▸c1456 R. Pecock Bk. Faith (Trin. Cambr.) (1909) 123 That feith..is thilke kinde or spice of knowyng, which a man gendrith and getith into his undirstonding.
1526 Tyndale Prol. Moses in Wks. 7 Fayth, is the beleuyng of Gods promises, and a sure trust in the goodnes and truth of God, which fayth iustified Abrah.
1581 J. Marbeck Bk. Notes & Common Places 375 Faith..maketh God & man friends.
1651 T. Hobbes Leviathan iii. xlii. 271 Faith is a gift of God, which Man can neither give, nor take away.
1690 J. Locke Ess. Humane Understanding iv. xviii. 348 Faith..is the Assent to any Proposition..upon the Credit of the Proposer, as coming immediately from God, which we call Revelation.
1744 Swift Serm. Trinity 52 Faith is an entire Dependence upon the Truth, the Power, the Justice, and the mercy of God.
1781 W. Cowper Expostulation 111 Faith, the root whence only can arise The graces of a life that wins the skies.
1835 Wordsworth Russ. Fugitive ii. xi, in Yarrow Revisited 132 That monumental grace Of Faith, which doth all passions tame That Reason should control.
1869 E. M. Goulburn Pursuit of Holiness iii. 21 Faith..the faculty by which we realize unseen things.
1921 A. Myerson Found. Personality ix. 167 Faith is really a transcendent Hope, renewing the feeling of energy.
1949 H. A. R. Gibb Mohammedanism vii. 113 They [sc. the Mu'tazilite movement] stressed the responsibility of the Believer as against the..emphasis on the sufficiency of faith, irrespective of ‘works’.
1951 W. Lewis Rotting Hill i. 4 Did it [sc. the decline of religion] rage beneath the surplice and eat away the roots of faith?
2002 Independent 7 Aug. 14/3 His refusal to ignore modern thought..led to his own crisis of faith.
2011 F. M. Jensen Study of Found. Justif. ix. 90 Faith is the instrument by which we are linked to Christ.
6.
a. A system of religious belief. Freq. with modifying word, as Christian, Jewish, Muslim, etc.
...
b. the faith (also the Faith): that system of religious belief which is regarded as true and correct; the true religion; spec. (esp. in earlier use) Christianity, or a particular branch of it. Also in extended use denoting non-religious beliefs (cf. sense A. 6d). Cf. defender of the faith at defender n. 2b. Sometimes without the in early use and in certain fixed phrases, as of faith: part and parcel of the faith. See also contrary to faith at Phrases 4a, Confession of Faith at confession n. 7a.
...
c. That which is believed, or required to be believed, on a particular subject; a belief. Also in pl.: points of faith, tenets. Cf. article of faith at article n. Phrases 1a. Now somewhat rare."
If we assume logic is not employing any sense of trust in its ability to reason and its conclusions, what number(s) can possibly be equal to themselves when divided by 10?
Why would we assume that? Logic
is the ability to trust in one's reasoning (or rather, it is trustful/proper reasoning).
