A Preface (that may not be necessary)
I share your passion for philosophical investigation of theological concepts that in turn is based on the Bible that in turn is based on God's revelation. That's the proper order: God -> Bible -> theology -> philosophy. Unfortunately most lay believers stops at Bible and do undisciplined "bring your own thoughts" into Bible verses and ignore the undercurrent of orthodox catholic (lower case "c") apostolic theology running throughout Christian History, a trend scholars name solo scriptura instead of sola scriptura (see this article).
But another mistake is to start with philosophy and similarly ignore the aforementioned orthodox catholic apostolic theology and bring in modern philosophy such as libertarian free will or "Pascal's wager" or secular understanding of "faith" divorced from right understanding of who Jesus is (who should be the center of any theology). Not that those concepts don't have a place in theological discussion, but they should not reframe orthodox theology using incompatible categories. Rather, those concepts should be purified and modified so they (as reflecting human psyche) are put AT THE SERVICE of orthodox theology, and in this we follow the example of St. Thomas Aquinas who tweaked Aristotelian concepts for his Biblical and orthodox theological system he summarized in the famous Summa Theologica. The relevancy of Aquinas theology today is concurrent with how well Aristotle understands the human psyche and human nature (which is gaining fresh appreciation in the 21st century) given the scientific advancement in psychology and neurology, which St. Thomas extended into the supernatural and eschatological future of humankind as revealed by God in the Bible. Think tanks like the Thomistic Institute carry on the Thomistic way of processing revelation and the world (including its scientific investigation) so philosophy remains theology's handmaiden.
So this answer tries to do philosophical investigation of "hope" as understood by the apostles; the understanding preserved for us in the Bible and THEN reflected upon by the Patristic, and later, medieval theologians such as St. Thomas Aquinas. Yes, there are several valid theologies of hope out there, some more orthodox than others, but I'm not a professional theologian / philosopher, so what I describe below is based on wide reading of various systematic theologies and philosophically flavored Thomistic books such as Josef Pieper's faith - hope - love, a collection of 3 philosophical essays on the 3 theological virtues according to Aquinas. I'm writing an SE answer intended only to whet the appetite of reading that book (so it's not a summary, just a teaser), which among the books I came across, remain the best philosophical investigation of the 3 virtues and has stood (somewhat) the test of time (in 1997 Ignatius Press reprinted the English translations from the original German, written in the 1960s).
Hope (Biblical & Theological)
In the few hundred years after the last canonical prophet Malachi (c. 432 BC, Nehemiah's 2nd visit to Jerusalem) and throughout the 2nd Temple period in which there were heroic acts such as the Maccabean Revolt and during which the fervent hope of a political Messiah who would deliver the nation of Israel into a golden age surpassing the old glory days of David and Solomon (1000-931 BC) was running high. Instead, Christianity redefined this national deliverance into an eschatological deliverance of the whole human race (all nations) from disorder, war, sin, injustice, and death by God through His incarnation in Jesus who currently lives forever guiding all people who are willing to be His disciples into an eschatological future of the new heaven and earth, the fulfillment of Eden (access to the Tree of Life is reopened and the ground is no longer cursed). This is a joyful place of endless exploration of God's goodness and greatness pursued by a new humanity (each person in a glorified body) who do life in the presence of God, that is, in peace, true wisdom, health, and love. If we ask a Christian: "what is the content of your hope", that would have been the answer.
God promised this and although we already see glimpses in the community of love that Jesus founded 2,000 years ago, recognizable when Jesus our Lord is present at the center of a community's life (his body ruled by his spiritual presence), Jesus has not yet come again in visible glory to inaugurate that eschatological future described above. So we live in a tension of having a foretaste but still yearning for complete fulfillment of God's promise. St. Paul described it in 1 Cor 13:12 ("For now we see only a reflection as in a mirror, but then face to face. Now I know in part, but then I will know fully, as I am fully known."). An important theme of the NT is encouragement in the midst of suffering as a theodicy to explain why persisting to keep this right desire for the content of that future (the virtue of hope) is no fool's errand but despite all appearances, it's bankrolled by the faithfulness of God who has been proven faithful before (partly by sending Moses in the OT and Jesus in the NT).
Hope (Philosophical)
Christian philosophy's role is to tie human soul's desire to this promise of the eschatological future (given by theology) and to re-kindle this desire despite tragedies, setbacks, wars, disappointments, depression, etc. Philosophy doesn't make light of these human challenges, but attempt to re-integrate these sufferings by re-setting our eyes on that future promise. So trust is needed. Trust in what? Well, in Christian context, it's obviously foolish to trust in the earthly goods that God has blessed us (even though they are good gifts that God is pleased to give His children), but trust in God the giver. The habitual propensity to trust in this way is faith, and the desire for God's future promise is hope. Philosophical analysis and investigation maintain the separation though they are obviously related.
How do we maintain this habitual stance to keep desiring God's promise? Aquinas (as philosophically processed by Josef Pieper) frame this as the theological virtue of hope where the grace infused into a human soul who is living in the tension described above (Chapter 1) can cure despair (Chapter 3) as well as can instill magnanimity and humility (Chapter 2) while staying away from presumption (Chapter 4) and from fraudulent imitation like Liberalism, Stoicism, and false fearlessness (Chapter 5).
Answering your questions
With the stage set above (proper biblical, theological, and philosophical context), I now proceed to give quick comments on your questions.
You are correct that "hope entails some degree of uncertainty ... but a strong desire for something to be true", the "something" being the "content of the hope" I described above.
Is it possible to redefine faith, traditionally rooted in strong beliefs, to encompass the prospect of being grounded in hope?
"faith" and "hope" are related, but as virtues (habits of the mind) they have different function: "faith" is about trusting, "hope" is about desire. Desire on the wrong promise is foolish, so shouldn't be named Christian hope but a pie in the sky. Christian hope is based on the true promise of God. "faith" then, is trusting the intention and the ability of this God to deliver that promise, which is often called "God's faithfulness". Who is the agent of Christian hope? Jesus Christ. So we have faith in Jesus to be our deliverer into the "promised land" of the new heaven and earth in a glorified body.
About "faith" and "hope"'s relationship to intellectual assent just ask yourself: what is the meaning of 'trust' and 'desire' in human relationship such as marriage? If I say "I desire to be happy with my wife" ("hope") and "I trust my wife's willingness to be happy together with me for the rest of my life" ("faith"), surely there IS intellectual assent involved?
The cognitive content of the "hope" is a picture of what "happy being with my wife" is; different husband has different picture. For the academically inclined, maybe the picture includes collaborating academically to publish a book. For those wanting grandchildren maybe the picture includes raising godly children who then produce godly grandchildren.
The cognitive content of the "faith" is accumulated proofs in the years of marriage as to whether my wife is keeping beside me when I'm sick, poor, or even (God forbid) incapacitated. If she seems to prefer to spend his time with other men, then this is a counterproof. If my wife has a different hope than myself, then although my wife has good character, our hopes are incompatible and thus by faith is misplaced.
Applying the same analysis to God,
The cognitive content of the "hope" is a theological processing of the narrative of revelation of God's intention as to the direction of history into the currently-only-partly-visible eschatological future.
The cognitive content of the "faith" is a minimal understanding of the gospel while the critical help to enable the cognitive faculty (i.e. reason) to assent to the gospel is the "light of faith" described in my other answer. In addition, "faith" as virtue (different in concept than "light of faith") is a habitual predisposition which you can read further in Josef Piper's essay on faith (not in the scope of this answer). Also, if the content of the "hope" is not what God promised (as recorded in the Bible understood correctly), these believers who DO have faith in Jesus Christ as deliverer will be disappointed as in victims of prosperity gospel who may have good Christian character but they were sold different cognitive content of the "hope" so more prone to despair and even deconversion!
CONCLUSION: I hope (desire) by separating the elements and recombining them with the help of Christian theology and Christian philosophy above enable you to see how saying "can faith be based on hope rather than belief or intellectual assent" is a confusing mash-up of terms that should be connected much differently:
- both faith and hope have their own cognitive content and that cognitive content need to be assented to by the intellect (reason). Otherwise faith and/or hope are irrational but humans are rational animals
- faith is belief that Jesus is our deliverer, yes, but then to the question "TO WHERE are we being delivered", the answer is: "the cognitive content of hope". I have seen examples of good Christians who are despairing (let's say a beloved child dies) and become depressed, but keeping their faith stoically in Jesus. The virtue to keep them joyful despite suffering is NOT stoicism (grit your teeth and move on) but hope. There is no joy without desire.
P.S.: Josef Pieper (German Catholic Thomist philosopher) is a very different figure than John Piper (American Calvinist theologian). I often got people confused when mentioning the former since the latter is much more famous, especially among evangelicals.
Follow-up comments
Revisiting your marriage analogy, your definitions of 'faith' and 'hope' presuppose that one is already convinced of the existence of one's spouse. However, in the way I phrased my question, I'm using hope with the more conventional meaning of "wishing that some fact be true", which entails uncertainty, but in your analogy it wouldn't make sense to say "I hope that my wife exists". If one is already convinced of one's spouse's existence, 'hoping' that one's spouse exists wouldn't make sense at all. If you already believe in and are utterly convinced of the existence of your wife, you wouldn't ever say that you 'hope' she exists.
You're correct that in my marriage analogy I'm already married with certain expectation (hope) with a little trust (faith of a mustard seed). My analogy presupposes the existence of God; it is meant to highlight the function of "hope" and "faith" in the journey in this life toward the uncertain future ("uncertain" psychologically speaking). I could have changed the analogy to my hope of finding a certain wife (who may not exist), but this would be a different analogy.
Remember how in the Preface I said that philosophy flows from theology? You can of course bring a word's common meaning into Christianity, but you must be willing for the word's meaning to be molded by theology; otherwise you'll be introducing a foreign category into Christian thoughts, thus misrepresenting Christianity.
In Christianity, hope is not simply wishful thinking, but tied to God's promise revealed earlier as the cognitive content. Prior to becoming Christian, yes, I can see how "faith" and "hope" is one package with God's existence. If God doesn't exist, then the faith and hope I described above go down along with the non-existence of God. On the other hand, let's say you believe in the existence of God, but as a Deist (watchmaker God). Then there is room for Christian "faith" and "hope" to follow when one is born again. I don't think the other way around is possible where simply because of the hope you can then believe the existence of God which is indeed wishful thinking. The arrow doesn't go in that direction.
But I shouldn't be too quick to dismiss that, because C.S. Lewis uses the argument from desire to show how our yearnings prove that there is a fulfillment beyond death. But it is a proof of the immortality of the soul; I don't think it's enough to guarantee the kind of God who is the basis for the full Christian "hope" (the goal) and "faith" (the way). Apologists usually use the Argument from Desire only to bolster the rationality of Christianity.
But once you become a Christian where existence of God is a necessary ingredient, "faith" and "hope" function differently in the spiritual life since they are cures to different spiritual maladies. The necessary healer is Jesus who give us grace for various spiritual illness, and that's why in Thomistic theologies "faith", "hope" and "love" are called theological virtues that grow stronger to help a disciple matures in different aspects.
What are your thoughts on the concept of 'hope-based faith' suggested by James Sennet in this lecture (6:54 to 10:30)?
Several take away points summarizing the segment: 1) she acts as if it is true until externally non-distinguishable from believers; 2) non-cognitive desire motivates her to do the act; 3) the gospel ideas being beautiful also feeds that desire; 4) doesn't say gospel is false, but just not enough evidence for. From that description, she is doing "fake it until it's real" hoping that one day the conviction will come. It is a "make believe" practice with an imaginary friend she hopes to be real.
From the Christian point of view, this is very laudable. The segment doesn't say, but I assume she also prays to God and starts a relationship with Jesus as prescribed by Christianity under the pretext that Jesus and the Father would reciprocate to her (by becoming real in her consciousness) according to what the Bible teaches Jesus and the Father would do. In other words, acting on the desire that this imaginary friend will turn up real, she acts her part.
Using my marriage analogy, this is similar to the wife (Carol) trying out a relationship with a husband (Bob, analogous to God) that she believes would reciprocate her "future picture" of a happy married couple life that Carol & Bob have worked out together (analogous to the cognitive content of hope described above). The fulfillment of that future picture depends on both sides doing their part (work hard, tangible support as lovers, selfless giving, etc.) seeing the construction of the pieces coming together (car, house, savings, kids, friends, etc.). But let's say Bob doesn't do his part (he is not as faithful to the project as Carol expected), or Carol found out that this "future picture" is no longer desirable to her (Carol abandons the project). Then this "future picture" in which Carol and Bob need to be existing in loving relationship proves to be illusory. They then separate. To Carol, this means Bob doesn't exist in her life anymore, analogous to God doesn't exist for Carol.
Of course, an atheist would say to Carol: "See, God doesn't exist". But a Christian with the light of faith would compassionately acknowledge that while the reality of God is not subjectively present in Carol's soul, he/she would pray together with her so someday the "light of faith" comes while encouraging Carol to keep on doing what she has been doing (there's no harm but only good, as the practice promotes a flourishing and satisfying life even without feeling God's presence). He/she would also rally up community support since loving community is the very picture that Jesus says already happened on this side of heaven. One powerful apologetics in the early church is "see how they love one another!".
Conclusion: If Carol doesn't yet have the conviction that the Biblical picture of hope is true (along with the prerequisite: existence of God), then she doesn't have the "light of faith". But since she "acts as if it is true", does Carol have the virtue of "faith"? I think while she is keeping it up she DOES, not by empirical evidence, but by the Bible teaching that God gives grace to those who ask for it, manifested in Carol's ability to overcome adversities that try to derail her trust in God (as powering her doxastic voluntarism without the light of faith) and her desire for heaven (which in this case IS the starting point).
In Christian theology the existence of God is a given and we trust what God promised in the Bible (the basis of this trust is a whole other discussion) but subjective cognition of God's presence and subjective conviction of His faithfulness can vary; it's great when a believer has it but it's not guaranteed.