In science, existence, or to "exist", as compared to non-existence (or "does not exist"), refers to something that has physical presence; an entity that has reality in the universe; something that has measurable properties; anything comprised of matter and or energy (see: mass-energy equivalence), fermions (matter) and or bosons (force), and vacuum or space—although, to note, there is still some debate, in the particle physics arena, as to whether or not a "true" or perfect vacuum can exist (see: nature abhors a vacuum).

Physical science
In physical science, there is a long history of assertion, proof, and or disproof of what exists and what does not exist.

In 1643, Italian physicist Evangelista Torricelli showed that ‘vacuums’ do exist, to a measurable degree, in contrast to the age old ‘nature abhors a vacuum’ dictum.

In 1798, Benjamin Thompson, via his cannon boring experiment, proved that ‘caloric’ or rather caloric particles—heat as a type of indestructible matter as espoused in the caloric theory or material theory of heat)—does not exist.

In 1875, Balfour Stewart and Peter Tait speculatively argued that ‘force’ is something that does not exist. [5]

In circa 1900, the energetics school, led by Wilhelm Ostwald, famously argues that ‘atoms’ do not exist; though Ostwald notably recanted his position in his last years.

In 1905, Albert Einstein famously proved that ‘either’ does not exist.

In 2013, Mexican-born American chemist Vicente Talanquer, in his critique of the use chemical teleology in US chemistry textbooks, asserted his view, in respect to the Le Chatelier's principle, that there is no ‘drive’ involved in chemical reactions—that drive, supposedly, is something that does not exist in nature: [6]

“The same type of thinking can be applied to the chemical processes in equilibrium. For example, when pressure is applied on a reacting system in chemical equilibrium, the system does not shift towards reactants or products "so that" or "in order to" counterbalance the effect of the change. What happens is that the change in pressure affects the probability of the forward and backward chemical processes going on in the system. The change in pressure affects the probability of collisions for the backward and forward processes in different ways. As a result, the likelihood of one process becomes higher than the other. There is NO DRIVE to attain a new equilibrium state. There is only particles randomly moving and interacting, involved in competing processes with different probabilities.”

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Human existence
French philosopher Rene Descartes, in his 1637 Discourse on Method, has given, invariably, the most-famous one-liner on the question of existence, specifically the subject of human existence, namely ‘I think therefore I am’, or in full: [1]

“Finally, since I thought that we could have all the same thoughts, while asleep, as we have while awake, although none of them is true at that time, I decided to pretend that nothing that ever entered my was any more true than the illusions of my dreams. But I noticed, immediately afterwards, that while I thus wished to think that everything was false, it was necessarily the case that I, who was thinking this, was something. When I noticed that this truth ‘I think, therefore I am’ was so firmly and certain that all the most extravagant assumptions of the skeptics were unable to shake it, I judged that I could accept it without scruple as the first principle of the philosophy for which I was searching. Then, when I was examining what I was, I realized that I could pretend that I had no body, and that there was no world nor any place in which I was present, but I could not pretend in the same way that I did not exist. On the contrary, from the very fact that I was thinking of doubting the truth of things, it followed very evidently and very certainly that I existed; whereas if I merely ceased to think, even if all the rest of what I had ever imagined were true, I would have no reason to believe that I existed. I knew from this that I was a substance, the whole essence or nature of which was to think and to which, in order to exist, has no need of any place and does not depend on anything material. Thus this self—that is, the soul by which I am what I am—is completely distinct from the body and is even easier to know that it, and even if the body did not exist the soul would be everything that it is.”

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Reason
The following is a noted 1976 quote on existence by English zoologist Richard Dawkins, from is his popular The Selfish Gene: “Intelligent life on a planet comes of age when it first works out the reason for its own existence.” Dawkins's point of view, here, to note, is near to the truth, but correctly, however, in the modern 21st century hmolscience + defunct theory of life perspective, according to life terminology upgrades, the statement should read:

"Intelligent [matter] on a planet comes of age when it first [calculates it own molecular formula] [and then] works out the reason [nature, or meaning] for its own existence.”

In other words, firstly, a human is a type of bound state intelligent matter (see: human molecule), so-to-speak, and not dead nor alive, as Dawkins would presume, i.e. in the famous 1915 words of Serbian-born American electrical engineer Nikola Tesla “there is no thing endowed with life”; secondly, that a hydrogen atom or a human molecule has a so-called "reason for existence" is a puzzling subject, to say the least, one that has not yet been worked out, as Dawkins seems to presume.

Quotes | Evolution
The following are other related scientific quotes on ‘exist’ or ‘existence’, specifically in the context of pre-evolution theories (e.g. metamorphosis), evolution theories (natural selection), or post-evolution theories (e.g. synthesis/analysis):

“Let us investigate more closely this property common to animal and plant, this power of producing its likeness, this chain of successive existences of individuals, which constitute the real existence of the species.”
— Buffon, “De la Reproduction en Generale et particuliere” (1749) [4]

“The primal plant is going to be the strangest creature in the world, which nature herself must envy me. With this model and the key to it, it will be possible to go on forever inventing plants and know that their existence is logical; that is to say, if they do not actually exist, they could, for they are not the shadowy phantoms of a vain imagination, but possess an inner necessity and truth. The same law will be applicable to all other living organism.”
— Johann Goethe, to Herder, 17 May 1787 [2]

“The struggle for existence amongst all organic beings throughout the world, which inevitably follows from their high geometrical powers of increase, will be treated of. This is the doctrine of Malthus, applied to the whole animal and vegetable kingdoms. As many more individuals of each species are born than can possibly survive; and as, consequently, there is frequently recurring struggle for existence, it follows that any being, if it vary however, slightly in any manner profitable to itself, under the complex and sometimes varying conditions of life, will have a better chance of surviving, and thus be naturally selected. From the strong principle of inheritance, any selected variety will tend to propagate its new and modified form.”
— Charles Darwin, The Origin of Species (1859)

Quotes | Other
The following are other related scientific quotes on ‘exist’ or ‘existence’:

“Being engaged lately in superintending the boring of cannon in the workshops of the military arsenal at Munich, I was struck with the very considerable degree of heat which a brass gun acquires in a short time in being bored, and with the still more intense heat (much greater than that of boiling water, as I found by experiment) of the metallic chips separated from it by the borer. The more I meditated on these phenomena, the more they appeared to me to be curious and interesting. A thorough investigation of them seemed even to bid fair to give a farther insight into the hidden nature of heat; and to enable us to form some reasonable conjectures respecting the existence, or non-existence, of an igneous fluid— a subject on which the opinions of philosophers have in all ages been much divided.”
— Benjamin Thompson, An Inquiry Concerning the Source of Heat which is Excited by Friction (1798)

“I moved swiftly, and rather superficially, from Feuerbach, to Darwin, to Nietzsche, to outline the process through which modern Western civilization came to define the meaning of existence in more humane terms than the unenlightened Chinese tradition had.”
— Fu Sinian (1919), reflection on Chen Duxiu’s “what is the point of human existence at all?” in the context of Confucian ideal of social service (Ѻ); cited by Jennifer Hecht (2003) in Doubt: a History (pg. 434)

“It has become a cheap intellectual pastime to contrast the infinitesimal pettiness of man with the vastnesses of the stellar universes. Yet all such comparisons are illicit. We cannot compare existence and meaning; they are disparate. The characteristic life of a man is itself the meaning of vast stretches of existences, and without it the latter have no value or significance. There is no common measure of physical existence and conscious experience because the latter is the only measure there is of the former. The significance of being, though not its existence, is the emotion it stirs, the thought it sustains.”
— John Dewey, Philosophy and Civilization (1931) [3]

References
1. (a) Descartes, Rene. (1637). Discourse on Method; in: Discourse on Method and Related Writings, trans. Desmond M. Clarke, Penguin edition (1999), part 4, 25.
(b) Bynum, W.F. and Porter, Roy. (2005). Oxford Dictionary of Scientific Quotations (exist: Goethe, 247:6, Descartes, 175:1, Anaximander, 12:1, Cran, 139:3; existence: Deway, 177:6, Buck, 98:3, Buff, 102:2, Berk, 56:1, Mill, 440:8, Dawk, 168:7, Darw, 156:8). Oxford University Press.
2. (a) Goethe, Johann. (1787). "Letter to Herder", May 17; in Italian Journey (1816-17), trans W.H. Auden and Elizabeth Mayer (1970), 310-11.
(b) Bynum, W.F. and Porter, Roy. (2005). Oxford Dictionary of Scientific Quotations (exist: Goethe, 247:6, Descartes, 175:1, Anaximander, 12:1, Cran, 139:3; existence: Dewey, 177:6, Buck, 98:3, Buff, 102:2, Berk, 56:1, Mill, 440:8, Dawk, 168:7, Darw, 156:8). Oxford University Press.
3. (a) Dewar, John. (1931). Philosophy and Civilization, reprinted in David Sidorsky (ed.), John Dewey: the Essential Writings (1977), 7.
(b) Bynum, W.F. and Porter, Roy. (2005). Oxford Dictionary of Scientific Quotations (exist: Goethe, 247:6, Descartes, 175:1, Anaximander, 12:1, Cran, 139:3; existence: Dewey, 177:6, Buck, 98:3, Buff, 102:2, Berk, 56:1, Mill, 440:8, Dawk, 168:7, Darw, 156:8). Oxford University Press.
4. Buffon. (1749). “De la Reproduction en Generale et particuliere”, Historie Naturelle, Generale et Particuliere, Avecla Description du Cabinete du Roi, Vol. 2, 18, trans. Phillip R. Sloan.
5. Stewart, Balfour and Tait, Peter G. (1875). The Unseen Universe: or Physical Speculations on a Future State (pg. #). Macmillan.
6. Talanquer, Vicente. (2013). "Email to Libb Thims", May 20.

External links
● Existence – Wikipedia.