This will be the first part of a two-part answer in conjunction with /u/hellcatfighter, with this part covering Tibet and Taiwan under the Qing and during the foundation of the Republic.
Tibet and Taiwan are two regions that had long been outside the control of rulers of China. The Mongol-established Yuan Dynasty did have some control over Tibet, but this was lost by the time of the Ming revolt, and preceding dynasties had otherwise had very little direct power projection in the region anyway. Taiwan, meanwhile, had a very limited Chinese presence (mostly a series of coastal fishing villages) at the time of the Qing conquest of China in the 1640s, and was still mainly indigenous territory, albeit with a small Dutch colony in what is now Tainan, and a Spanish colony in what is now the northern part of Taipei.
However, by the end of the eighteenth and seventeenth centuries, respectively, the two regions had come under direct Qing control as a result of their military expansion. Going chronologically, Taiwan, which had been used as a base by refugee Ming loyalists since their expulsion of the Dutch in 1662, was conquered in 1685 following the suppression of the Three Feudatories Revolt (which the Taiwanese rebels had supported); Tibet was made into a de facto protectorate following the expulsion of a Zunghar occupation force in 1720, and Tibetan political autonomy was severely curtailed after the conclusion of the Qing-Nepal wars (the Tibetan lamas were blamed for inciting a Nepalese invasion) in 1792. The latter incident already shows that the Qing considered Tibet an integral part of the Qing empire, but that is not necessarily the same as seeing it as an integral part of 'China', in the sense of a territorial unit unto itself. Rather, Tibet was one of a series of territorial units under Qing control through a distinct system of administration and cultural relations. Taiwan, on the other hand, was considered a part of 'China' in particular, being made a sub-provincial unit of Fujian and under Qing regular administration (albeit with concessions to indigenous groups). During the latter stages of the Sino-French War of 1884-5, the risk of losing Taiwan as a result of the French occupation of the Pescadores (which allowed them to intercept Qing supply routes to the island) led to the Qing suing for peace and giving up their interests in Vietnam in concession, and subsequently establishing Taiwan as its own province of China.
By the time of the 1911 Revolution, however, the question of whether Taiwan constituted an integral part of Chinese territory was somewhat immaterial, as it had been annexed by Japan as a result of the Treaty of Shimonoseki in 1895. But Tibet of course remained part of the Qing Empire, and so its fate was brought up in the question of what form the new, post-imperial China would take. This was particularly pressing because the core rhetoric of the revolutionary forces had revolved around matters of ethnic identity: many presented the revolt as specifically against the Manchus and a movement to 'revive' the (recently constructed) Han Chinese nation. But the consequence of that was that the post-imperial state might not be able to legitimately claim control over the vast Qing peripheries. In any case, two of these peripheries, Mongolia and Tibet, the former of which saw its loyalty as being to the Qing imperial house and the latter of which sought outright independence, successfully declared independence from China as the Aisin Gioro house was deposed.
The question of Tibet was not a new one. Radical reformists like Liang Qichao, who after 1898 championed the idea of a Han-led constitutional monarchy while preserving the Qing ruling house, came to argue for the retention of the Qing empire's borders as they then existed, despite their aiming to promote the Han specifically. But this was not a universal position. Zhang Binglin, another radical agitator, called for the breakup of the empire on the basis that Han and non-Han should be prevented from intermixing. The nationalist republicans, like Sun Yat-Sen, did not necessarily plan that far ahead, but their rhetoric of Han nationalism suggests at least that the continuation of 'Chinese' control over the Qing empire's non-Chinese constituents would not be maintained through accommodation, if at all. Nevertheless, despite the republican character of the revolution that would overthrow the Qing in 1912, it was Liang Qichao's ideal of a Chinese national empire that won out, if nothing else because the declarations of Mongolian and Tibetan independence threatened to strip even more regions away from the new Republic, and made clear the extent of potential territory that would be 'lost' if the new China were a narrowly nationalist one. The adoption of a five-striped flag to represent the ethnic unity of the new state would prove to be the key symbolic move in the claim to the territories inhabited by the peoples depicted, and a critical point in the development of Chinese nationalism from one purely of Han nationalism towards one that sought to encompass a wider range of peoples. However, in the event, the new Republic never had the strength to enforce such claims.
But it is worth adding that there were contingent factors at work in creating the sense of a multiethnic Chinese nation, at least at the level of state ideology. Many of the revolution's leading figures, especially Li Yuanhong, were explicitly of the Han-nationalist sort, and his view was for the establishment of a Republic in the eighteen provinces of China proper and the expulsion of the Manchus beyond its northern border, demarcated by the Great Wall. Old ideologues like Sun Yat-Sen took a more measured approach in proposing to retain all of the Qing Empire's territories but establishing a clearly Han-dominant political consensus. But neither of these men held the position of President during the critical period of the Republic's initial consolidation. Yuan Shikai, the compromise candidate who was elected President on 14 February 1912, two days after the abdication of the Aisin Gioro house, had been a committed monarchist, and established a new Republic that, critically, embraced a vision of a plural state that could potentially reintegrate Tibet and Mongolia, with policies intended to accommodate Tibetan, Mongolian and Muslim practices, and with seats prepared for Tibetan and Mongolian delegations at the National Assembly should they choose to send them. These gestures did not end up seeing the regions actually reintegrated during Yuan's tenure, nor indeed that of any president of the Republic of China, but it can be said that attempts to legitimise continued claims to Qing territory have been a feature of post-imperial states since the moment the empire ended.
Sources, Notes and References
Well China (and by that we mean the People's Republic of China, i.e. China, and not the Republic of China, i.e. Taiwan) does own Tibet by any reasonable standard of international diplomacy and sovreignty. Even Tibetans would agree with the definition, their main issue being the word "own" which would denote ownership. This would imply that if someone steals something, the owner is still the original possessor of the object, the same principal (Tibetans would tend to argue) applies to countries. Tibet is owned by Tibetans, China stole it, however Tibet is still owned by Tibetans even if China maintains possession.
Of course, we've zoomed right past history and to the competing philosophical traditions of post-Westphalian Statehood and the Sino-Centric View of the world. I would encourage you to look into the sources listed below as what follows below will be a rather short summary.
The Sino-Centric view of the world is often described as "Confucian" which portrays the world as a series of concentric circles starting with the Emperor, or Son of Heaven, at the very center, his immediate family around him, then his extended family, then the provinces of China, then the vassals of China, then the "civilized" or "pacified" barbarians, then the "uncivilized" and "unpacified" barbarians. In this perspective, the spheres of influence expand ever outward, and Chinese history is (for better or for worse) often viewed through this perspective. At the very beginning of Chinese Dynastic history (approximately 2500 years ago) China was a collection of Warring States, and it was the Qin (from where we get the Latinized Sin- and the English China) who were at the edge of the "Chinese" sphere who ultimately conquered, united, and began the Dynastic period of Chinese history. This all occurred in the "Cradle" of Chinese civilization, the Huang River Valley. The next step in the evolution of Chinese history was the conquest of the Yangtze River Valley (the state of Chu), encompassing what we call the "Core" of Chinese civilization. This, from the outside, seems like a seamless assimilation - Shanghai, Hong Kong, Wenzhou, Canton, Guangzhou, all seem like inarguably Chinese places from the outside. Yet local cultures still persist and a north-south cultural (and linguistic) divide persists to the present day and is unlikely to diverge any time soon.
The next phase of Chinese history was to consolidate the Core holdings (this Cradle/Core dynamic is illustrated in cartographic form in John Keay's China: A History though I don't know if he's the originator of the idea) and expand into the barbarian zones. This was most successful in Korea and Vietnam, with lesser successes in Japan, the Ryukyu Islands, and Taiwan (though obviously Taiwan's 20th Century history has diverged very much from most of its historic trajectory). Korea, particularly, was so deeply assimilated into the Chinese Confucian worldview that they took pride in their status as the "Younger Brother" of China. In the Confucian worldview, the world would be set right by the establishment of five relations, one of them being younger brother to older brother. See here for more details. Vietnam and Japan also assimilated not just a Chinese form of Buddhism, but also Taoist and Confucian ideas. Korea made Confucianism their state religion starting in 1392 with the rise of the Yi Dynasty, and Japan followed suit with a state-level form of Confucianism in 1605.
Chinese cultural expansion west, however, was a very different story. Most maps of China throughout the ages show that a panhandle forms in what is today Gansu Province. Often referred to as the "Gansu Corridor" to the north of Gansu is the Mongolian Plateau and to the south is the Tibetan Plateau. This corridor was China's land connection to the Silk Route, established in the Han Dynasty to encourage trade with first India, and later the Middle East and Europe. Establishment of this trade route, where Chinese intellectual life once thought they were essentially alone in a realm of expanding spheres, has been compared to the European discovery of the new world. For the first time, Chinese intellectuals had to grapple with the fact that there were people who existed beyond even the farthest reaches of the "uncivilized" or "uncultured" barbarians, who had Chinese silks, but no knowledge of the Son of Heaven.
While I'm sure there is some room for nuance, essentially the Chinese worldview extended the zone of the uncivilized and unpacified barbarians to essentially encompass the whole world outside of China and her vassals. How China dealt with this over the centuries has waxed to wars of conquest and sea-voyages of exploration, to completely focusing internally and retracting from the world, not concerning themselves with whether the barbarians (Japan particularly) were submitting, kowtowing, or sending the tribute that kept the Chinese Vassal system going.
The Tibetans throughout their history have, of course, had a lot of interaction with China, but were almost always in the zone of the uncivilized and unpacified barbarians. In the eighth century, the Tibetans held a debate between the two competing traditions of Buddhism in their Empire: between the Indian Vajrayana tradition, and the Chinese Mahayana tradition. The results of the debate are unclear, but the traditional retelling is that the Vajrayanists won (and the Nyingma school of Buddhism which claims direct descent from the winners of this debate, have also incorporated certain teachings and practices that they claim descend from the Mahayanists as well). The Tibetans have also always had their own indigenous traditions of rulership, governance, and laws. The first time the Tibetans came under, ahem, Chinese overlordship was during the Yuan Dynasty when the Sakya hierarch Sakya Pandita submitted voluntarily to Godan Khan, a relative of the famous Mongol Emperor Khubilai Khan. Sakya Pandita's nephew Chogyal Phagspa later established a formal relationship (called awkwardly in English "priest-patron") with Khubilai Khan, and ruled Tibet as a vassal to the Mongols. While the Mongols were busy fighting a losing battle against the Ming rebels, the Sakya had their own revolt in the form of Jangchub Gyaltshen of the Phagmodrupa dynasty. The Phagmodrupa "Kings" fell to the Rinpungpa, who fell to the Tsangpa, who fell to a distant descendant of Khubilai Khan, Gushri Khan, who handed over his conquered Tibet to his teacher, the Fifth Dalai Lama. The Fifth Dalai Lama in an attempt to pacify the relationship between the Khoshut Mongols and the Manchu Qing Dynasty, made an informal relationship with the Shunzhi Emperor, establishing a temporary peace between his Tibet, his patron Gushri Khan, and his new student, the Emperor of the Qing Dynasty. Over the course of the 1700s, see here, here, and here for details, the Qing Dynasty (which were ethnically Manchu, not Chinese) gained more and more control over Tibet, ruling them as a vassal apart and separate from China and the Chinese people.
When the Qing Dynasty fell to the Xinhai Revolution in 1911, Tibet had its own revolution. So 1912 saw the establishment of the early Chinese Republic and the simultaneous establishment of the independent Tibetan State.
Of course, the Republic of China didn't have a new Emperor (let's just ignore Yuan Shikai's failed bid for imperialship for now) and this required a new framework of thought for China's place in the world. The Opium Wars, the Boxer Rebellion, China's Open Door Policy, and all of the humiliations of the 1800s (while the Chinese people themselves remained in a subservient position to a foreign dynasty anyway) all had their cultural and long-term impact on the Chinese psyche. Chinese (as well as others, Japanese, Koreans, Vietnamese, etc.) had started leaving their homeland and were being educated in Western countries, in Western languages, and assimilating Western ideas. Like republicanism. The establishment of the new Chinese republic seemed to be where China was headed, but old ways die hard, and a 2400 year old tradition of centralized autocracy was a hard nut to crack (and some might argue quite successfully that China never did crack it). The Republic almost immediately gave way to a brutal period of Warlordship where men who could amass enough weapons (often with outside support) ruled their own province. Bandits turned into marauding bands of armies (where the average age was 14). And even what was left of the Republic first nearly descended into a new Empire with Yuan Shikai, but then turned into a German style autocracy under Chiang Kai-shek.
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