Andrew Carnegie was (or at least represented himself to be) a paragon of the "rags to riches" story of American individualism, someone who started out as a poor immigrant and became a "self-made man." This is at least partly myth and exaggeration, since Carnegie was never poor, per se, but started out in a modest, "skilled labor" job as a telegrapher.
Nevertheless, his person has contributed greatly to this view of American individualism: that hard work and education can turn a working person into a millionaire and even put them in a position to help others. Carnegie's "second career" as a philanthropist and his article "the Gospel of Wealth" proves that he did believe that individuals have some responsibility to better their communities to the extent that they are able to do so. However, this must, on the view he took, be strictly voluntary.
William Jennings Bryan's views were similar to Carnegie's in many ways. He, too, believed in hard work and the possibility of an individual lifting themselves up and giving their children a better life than the one that they had. His principle difference with Carnegie is that Bryan believed that society owed it to the individual to make this possible.
American capitalism had, by Bryan's day, reached extremes which made it difficult if not impossible for working people to better themselves simply through intelligence and hard work. If this was to change, it required not just the charity of the wealthy, but structural, political change. The following lines from Bryan's famous "cross of gold" speech provide evidence for this.
When you come before us and tell us that we shall disturb your business interests, we reply that you have disturbed our business interests by your action. We say to you that you have made too limited in its application the definition of a businessman. The man who is employed for wages is as much a businessman as his employer. The attorney in a country town is as much a businessman as the corporation counsel in a great metropolis. The merchant at the crossroads store is as much a businessman as the merchant of New York. The farmer who goes forth in the morning and toils all day, begins in the spring and toils all summer, and by the application of brain and muscle to the natural resources of this country creates wealth, is as much a businessman as the man who goes upon the Board of Trade and bets upon the price of grain. The miners who go 1,000 feet into the earth or climb 2,000 feet upon the cliffs and bring forth from their hiding places the precious metals to be poured in the channels of trade are as much businessmen as the few financial magnates who in a backroom corner the money of the world.
We come to speak for this broader class of businessmen. Ah. my friends, we say not one word against those who live upon the Atlantic Coast; but those hardy pioneers who braved all the dangers of the wilderness, who have made the desert to blossom as the rose—those pioneers away out there, rearing their children near to nature’s heart, where they can mingle their voices with the voices of the birds—out there where they have erected schoolhouses for the education of their children and churches where they praise their Creator, and the cemeteries where sleep the ashes of their dead—are as deserving of the consideration of this party as any people in this country.
It is for these that we speak. We do not come as aggressors. Our war is not a war of conquest. We are fighting in the defense of our homes, our families, and posterity. We have petitioned, and our petitions have been scorned. We have entreated, and our entreaties have been disregarded. We have begged, and they have mocked when our calamity came.
We beg no longer; we entreat no more; we petition no more. We defy them!
Bryan was as much an individualist as Carnegie. Although he was a champion of poor and working-class people, he did not aim to abolish the American ideal of individual property ownership. Rather, he sought to make that ideal more accessible to a wider range of individuals through political reform.
Roosevelt took, in many ways, the opposite tack. While Carnegie and Bryan both sought to find ways for the individual to actualize himself within the existing social order (Carnegie through hard work and philanthropy, Bryan through political reform), Roosevelt's vision of individualism was, for the most part, centered on places outside of normal society: in war, for example, or in the "great outdoors," where individuals would prove themselves either on the battlefield or in the wilderness. Roosevelt's image as a "man's man," as the soldier who had charged up San Juan hill or the big game hunter who carried back trophies from safari, was a big part of his political appeal. For him, individualism was less a way of getting on in society than it was the ability to get by outside of society entirely and then to bring this "rugged individualism" back into the realm of politics and economics.